Performance analysis of Vehicular Platoon
considering V2V Communication
基于车车通信的车辆队列性能分析
Keqiang LI 李克强
State Key Lab. of Automotive Safety and Energy
汽车安全与节能国家重点实验室
Tsinghua University 清华大学
1
Outline
Background
Problem Statements
Performance Analysis considering V2V
communication
Conclusions
State Key Lab of Automotive Safety and Energy· Tsinghua Univ.
2
Outline
Background
Problem Statements
Performance Analysis considering V2V
communication
Conclusions
State Key Lab of Automotive Safety and Energy· Tsinghua Univ.
3
Background
 History of Adaptive Cruise Control (ACC）
 Bosch Ltd, developed the first prototyping ACC
 Mercedes Benz first to offer ACC, called Distronic system
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4
Background
 Next Generation of ACC
 V2V/V2I communication
 Multi-vehicle cooperative control (e.g.,
Vehicular Platoon, CACC)
 Potential Benefits
Potential
Benefits
ACC
 Short following distance for better traffic
capacity
 Faster reaction to leader fluctuation to
reduce potential congestion
 Reduced aerodynamic resistance for
better fuel economy and less emission
 More reliable automation for comfortable
mobility
Economic
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5
Background
 Platoon: Topics and Researchers
 Research topics
1) Selection of spacing policies; 2) Communication delay; 3) Vehicle dynamic
uncertainty; etc.
 Researchers
1) U.S.: J. Hedrick, P. Seiler, S. Darbha, Huei Peng etc.
2) Europe: Jeroen Ploeg, N. van de Wouw, H. Nijmeijer, etc.
3) China: Keqiang Li, Feng Gao, etc.
 Platoon: Experiments
US: PATH
Europe: SARTRE
Japan: Energy ITS
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6
Background
 Platoon: Emerging Topics
Different Information Flow Topologies
Typical Information Flow
...
(a)
...
...
(b)
Connected Vehicle by V2V
(c)
...
(d)
...
(e)
...
(f)
N
N-1
N-2
 Platoon: Key Tasks
2
1
0
...
1. Model and design of a platoon system with a broad types of topologies.
2. Performance analysis of different topologies on platoons.
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7
Outline
Background
Problem Statements
Performance Analysis considering V2V
communication
Conclusions
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8
Problem Statements

Modeling of Platoons from the viewpoint of Networks of Dynamical Systems
 From Control Perspective
1. Dynamics + Communication
2. Control Theory + Graph Theory
 Research topics
1. Dynamic: single integrator,
double integrator, linear
dynamic, nonlinear dynamic

Applications
2. Communication: data rate,
switching topology, time-delay
A vehicular platoon can be viewed as a one-dimensional network of dynamical system
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9
Problem Statements

Modeling of Platoons under the Four-component Framework
Information Flow Topology
Distributed Controller
Controller Ci
CN
uN
ui
...
N
node i
Vehicle Dynamics
C1
Ci-1
dr,i
ddes,i
u1 v0
ui-1
t
...
i-1
LV
1
0
Formation Geometry
 Vehicle Dynamics: The vehicle dynamics describe the behavior of each node;
 Formation Geometry: The formation geometry dictates the desired distance between any
two successive nodes.
 Distributed Controller: The distributed controller implements feedback control for each
vehicle;
 Information Flow Topology: The information flow topology defines how nodes exchange
information with each other;
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Problem Statements

Model for Vehicle Longitudinal Dynamics

Linear Dynamics
𝑥𝑖 𝑡 = 𝐴𝑥𝑖 𝑡 + 𝐵𝑢𝑖 𝑡
0
𝑝𝑖
0
𝑥𝑖 𝑡 = 𝑣𝑖 , 𝐴 =
𝑎𝑖
0
0
0
1
0
,
𝐵
=
1
1
0 −
𝜏
𝜏
 Feedback linearization technique is used to convert the nonlinear model into
a linear one;
 The vehicle dynamics is assumed to be homogeneous.

1
0
Model for Formation Geometry
&lim 𝑣𝑖 𝑡 − 𝑣0 𝑡
𝑡→∞
= 0,
𝑖 = 1,2, ⋯ 𝑁
&lim 𝑝𝑖−1 𝑡 − 𝑝𝑖 𝑡 − 𝑑𝑖−1,𝑖 = 0
,&
𝑡→∞
 Constant distance (CD) policy
 Constant time headway (CTH) policy
 Nonlinear distance policy
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Problem Statements

Model for information flow topology
...
The V2V communication can generates
various information flow topologies.
(a)
a) Predecessor following topology;
 PF topology
...
...
(b)
b) Predecessor-leader following topology;
 PLF topology
(c)
c) Bidirectional topology;
 BD topology
...
(d)
d) Bidirectional-leader topology;
 BDL topology
...
(e)
e) Two predecessors following topology
 TPF topology
f) Two predecessor-leader following topology
 TPLF topology
...
(f)
N
N-1
N-2
2
1
0
...
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12
Problem Statements

Model for information flow topology
 Algebraic Graph Theory
 Viewed as a directed graph G, and use Pinning matrix, Adjacent matrix and
Laplacian matrix to model the connections.
 The communication is assumed as to be perfect. There is no delay, data loss etc.
 Definitions
• Pinning Matrix
• Adjacent Matrix
To model the information
flow from the leader to
followers
To model the information
flow among followers
𝒫=
𝓅1
• Laplacian Matrix
An induced matrix from
adjacent matrix
ℒ = 𝑙𝑖𝑗 ∈ ℝ𝑁×𝑁
&𝒜𝑁 = 𝑎𝑖𝑗 ∈ ℝ𝑁×𝑁
⋱
𝓅𝑁
&𝓅𝑖 = 1 , 𝑖𝑓&& 𝛼0 , 𝛼𝑖 ∈ 𝐸
&𝑎𝑖𝑗 = 1, 𝑖𝑓&& 𝛼𝑗 , 𝛼𝑖 ∈ 𝐸
&𝑎𝑖𝑗 = 0, 𝑖𝑓&& 𝛼𝑗 , 𝛼𝑖 ∉ 𝐸
𝑙𝑖𝑗 =
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&−𝑎𝑖𝑗 &&&&&&&, &&𝑖 ≠ 𝑗
&
𝑁
𝑘=1 𝑎𝑖𝑘
, &𝑖 = 𝑗
.
13
Problem Statements

Model for information flow topology
 Example 1: Bidirectional Topology
1
0
𝒫=
⋱
0
&𝒜𝑁 =
0
1
3
1
0
1
1
0
⋱
。。。
N
⋱
⋱
1
1
0
1 2 3 。。。 N
1
2
ℒ=
1
−1
−1
2
−1
−1
2
⋱
⋱
⋱
−1
−1
1
⋱
−1
1
 Example 2: Predecessor-leader following Topology
N-1
N
N-2
1
0
...
1
𝒫=
1
&𝒜𝑁 =
⋱
1
0
1
0
1
0
⋱
ℒ=
⋱
1
0
−1
0
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1
−1
1
⋱
14
Problem Statements
Model for Distributed Controller


Linear State Feedback Controller
𝑢𝑖 𝑡 = −
𝑘𝑝 𝑝𝑖 − 𝑝𝑗 − 𝑑𝑖,𝑗 + 𝑘𝑣 𝑣𝑖 − 𝑣𝑗 + 𝑘𝑎 𝑎𝑖 − 𝑎𝑗
𝑗∈&𝕀𝑖
 The local controller in node 𝑖 only uses its neighborhood information specified
by&𝕀𝑖 .
 The controller is assumed to be linear for the convenience on theoretical analysis.
 The controller in each node is assumed to be homogeneous.
Formulation for the Closed-loop Dynamics of Platoons

 Tracking error 𝑥𝑖 𝑡 = 𝑥𝑖 𝑡 − 𝑥0 𝑡 − 𝑑𝑖 , &&& 𝑑𝑖 = 𝑑0,𝑖 , 0,0


 Collective state vector 𝑋 = 𝑥1𝑇 , 𝑥2𝑇 , ⋯ , 𝑥𝑁𝑇
𝑇
∈ ℝ3𝑁×1 ,
 Collective input vector 𝑈 = 𝑢1 , 𝑢2 , ⋯ , 𝑢𝑁
𝑇
∈ ℝ𝑁×1 ,
Controller
𝑈 = − ℒ + 𝒫 ⊗ 𝑘𝑇 ∙ 𝑋
Closed-loop Dynamics of platoon
𝑇
𝑇
𝑘 = 𝑘𝑝 , 𝑘𝑣 , 𝑘𝑎 &,⊗&is Kronecker product.
𝑋 = 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 ∙ 𝑋
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Problem Statements

Unified Closed-loop Dynamics of Platoons
𝑋 = 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 ∙ 𝑋
Information Flow Topology
Distributed Controller
Controller Ci
CN
uN
ui
...
N
node i
Vehicle Dynamics

C1
Ci-1
dr,i
ddes,i
u1 v0
ui-1
t
...
i-1
LV
1
0
Formation Geometry
Questions
Q1. What’&the&stabilizing&region&of&controller&gain&𝑘 under different information
flow topologies?
Q2. How to choose topology and design controller to improve stability?
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Outline
Background
Problem Statements
Performance Analysis considering V2V
communication
Conclusions
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Performance Analysis considering V2V communication

Performance Definition
 Closed-loop Stability: A platoon with linear time-invariant dynamics is said to be
closed-loop stable if and only if the closed-loop system has eigenvalues with strictly
negative real parts
 Stability Margin: The stability margin of a platoon is defined as the absolute value of the
real part of the least stable eigenvalue, which characterizes the convergence speed.
Im
Stability Margin
dmin
R
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Performance Analysis considering V2V communication

1. Stability Region Analysis
 Unified Closed-loop Dynamics for vehicular platoons
𝑋 = 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 𝑋
𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 ∈ ℝ3𝑁×3𝑁
 Requirements of Closed-loop Stability
Re 𝜎𝑖 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘
𝑇
 large-scale matrix, which is
difficult to analyze directly.
<0
 Dynamic decouple by using similarity transformation
 Key step: to decompose the large-scale vehicular platoon into multiple subsystems,
which is easier to handle the closed-loop stability
𝑁
𝑆 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 = &&&&
𝑆 𝐴 − 𝜆𝑖 𝐵𝑘 𝑇
𝑖=1
𝑆 ∙ is the spectrum of a matrix.
𝜆𝑖 is the eigenvalue of ℒ + 𝒫
 Multiple small-scale matrix, whose size
is equal to that of node dynamics (n=3).
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Performance Analysis considering V2V communication

1. Stability Region Analysis
 Dynamic decouple by using similarity transformation
𝑁
𝑆 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 =
𝑆 𝐴 − 𝜆𝑖 𝐵𝑘 𝑇
𝑖=1
 Routh–Hurwitz stability criterion
𝑠𝐼 − 𝐴 − 𝜆𝑖 𝐵𝑘 𝑇
s3&
s2
𝑠1
𝑠0
1
𝜆𝑖 𝑘𝑎 + 1
𝜏
𝜆𝑖 𝑘𝑣 𝜆𝑖 𝑘𝑎 + 1 − 𝜆𝑖 𝑘𝑝 𝜏
𝜏 𝜆𝑖 𝑘𝑎 + 1
𝜆𝑖 𝑘𝑝
𝜏
= 𝑠3 +
𝜆𝑖 𝑘𝑣
𝜏
𝜆𝑖 𝑘𝑝
𝜏
𝜆𝑖 𝑘𝑎 +1 2
𝑠
𝜏
+
𝜆𝑖 𝑘𝑣
𝑠
𝜏
+
𝜆𝑖 𝑘𝑝
𝜏
.
 Real coefficient polynomials.
 It needs 𝜆𝑖 to be positive real
number.
&𝑝>0
𝑘
&𝑘𝑣 > 𝑘𝑝 𝜏 𝜆𝑖 𝑘𝑎 + 1
&𝑘𝑎 > − 1 𝜆𝑖
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20
Performance Analysis considering V2V communication
1. Stability Region Analysis

Consider a homogeneous platoon with linear controllers given by
𝑋 = 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 𝑋
(1.1) If graph&𝐺 satisfies certain conditions (all the eigenvalues of ℒ + 𝒫&are positive
real numbers), the platoon is asymptotically stable if and only if
&𝑝>0
𝑘
&𝑘𝑣 > 𝑘𝑝 𝜏 min 𝜆𝑖 𝑘𝑎 + 1
&𝑘𝑎 > − 1 max 𝜆𝑖

Remarks
 𝜆𝑖 is the eigenvalue of ℒ + 𝒫 (𝜆𝑖 &need to be positive real number).
 This result can cover a lot of information flow topologies, including all the
aforementioned topologies
 The influence of information flow topology on stability is mainly reflected by 𝜆𝑖 .
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Performance Analysis considering V2V communication

1. Stability Region Analysis– Simulation Results
𝑘𝑝 = 1, 𝑘𝑣 = 2, 𝑘𝑎 = 1, 𝜏=0.5
(a)
(d)
(b)
(e)
&𝑝>0
𝑘
&𝑘𝑣 > 𝑘𝑝 𝜏 min 𝜆𝑖 𝑘𝑎 + 1
&𝑘𝑎 > − 1 max 𝜆𝑖
Stable
(c)
(f)
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Performance Analysis considering V2V communication

1. Stability Region Analysis– Simulation Results
𝑘𝑝 = 1, 𝑘𝑣 = 0.2, 𝑘𝑎 = 1, 𝜏=0.5
(a)
(d)
(b)
(e)
&𝑝>0
𝑘
&𝑘𝑣 > 𝑘𝑝 𝜏 min 𝜆𝑖 𝑘𝑎 + 1 Unstable
&𝑘𝑎 > − 1 max 𝜆𝑖
(c)
(f)
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Performance Analysis considering V2V communication

2. Scaling trend of Stability Margin with increasing platoon size
Consider a homogeneous platoon with linear controllers given by
𝑋 = 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 𝑋
(2.1) if the graph 𝐺 is in Bidirectional topology, then the stability margin of
platoons decays to zero as 𝑂 1 𝑁 2
(2.2) if the graph 𝐺 is in BDL topology, then the stability margin of platoons is
always bounded away from zero.
N
N-1
N-2
1
...
0
 Decay to zero with increasing size;
 Independent with controller gains.
Bidirectional (BD) topology
N
N-1
N-2
1
...
0
 Will not decay to zero as
platoon size increases.
Bidirectional-leader (BDL) topology
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Performance Analysis considering V2V communication

2. Scaling trend of Stability Margin with increasing platoon size - Proof
Step. 1 Closed-loop Dynamics
𝑋 = 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 𝑋
Step. 2 Similarity transformation
𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 ∈ ℝ3𝑁×3𝑁
𝑁
𝑆 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 =
𝑖=1
Step. 3 Eigenvalue Analysis
𝑠𝐼 − 𝐴 − 𝜆𝑖 𝐵𝑘 𝑇
Im
𝑆 𝐴 − 𝜆𝑖 𝐵𝑘 𝑇
= 𝑠3 +
𝜆𝑖 𝑘𝑎 +1 2
𝑠
𝜏
+
𝜆𝑖 𝑘𝑣
𝑠
𝜏
Stability Margin
+
𝜆𝑖 𝑘𝑝
𝜏
.
dmin
R
Stability Margin
dmin = Re smin
= O 𝜆min .
 For Bidirectional topology
dmin = O 𝜆min = 𝑂 1 𝑁 2
 For Bidirectional-leader
topology
dmin = O 𝜆min =constant number
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decay to zero
Independent
with size
25
Performance Analysis considering V2V communication
2. Scaling trend of Stability Margin with increasing platoon size - Simulations
N-1
N
N-2
1
0
...
𝑘𝑝 = 1, 𝑘𝑣 = 2, 𝑘𝑎 = 1, 𝜏=0.5
Bidirectional topology
0
10
0
10
-1
10
-1
Stability Margin
10
Stability Margin

-2
10
-2
10
-3
10
-3
10
-4
10
-4
10
-5
10
10
100
The number of vehicles:N
-5
10
10
100
The number of vehicles:N
500
500
N
N-1
N-2
1
0
...
Bidirectional-leader topology
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26
Performance Analysis considering V2V communication

3. Stability Margin Improvement – Topology Selection
Consider a platoon with linear controllers given by
𝑋 = 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 𝑋
(3.1) if the graph 𝐺 is undirected, to maintain bounded stability margin, it needs at least
lots of followers (i.e. Ω(N) = O(N)) to obtain the leader’s information.
Remarks
i+2
i
i+1
i-1
i-h
i-2
...

...
...
i+h
...
 BD&topology&is&a&special&case,&i.e.,&Ω(N)=1, for which the stability margin decays to
zero as the platoon size increases;
N
N-1
N-2
1
...
0
 It implies that the information flow from the leader is more important than that
among the followers.
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Performance Analysis considering V2V communication
3. Stability Margin Improvement – Topology Selection

 To maintain bounded stability margin, the tree depth of graph G should be a constant
number and independent of the platoon size N.
Tree dpeth&𝑐 = max 𝑛1 , 𝑛2 − 𝑛1 , ⋯ , 𝑛𝑝 − 𝑛𝑝−1 , 𝑁 − 𝑛𝑝 + 1 , where& 𝑛1 , 𝑛2 , ⋯ , 𝑛𝑝 , 1 ≤ 𝑛1 ≤
•
⋯ ≤ 𝑛𝑝 ≤ 𝑁&is&the&set&of&followers&pinned&to&the&leader
...
n+1
n
2
1
n
0
...
N=c(k+1)
...
...
ck+1
c(k-1)+1
ck
...
c
1
0
c
0
N=2n
 Tree depth is N/2, increasing with
platoon size;
 Stability margin decay to zero;
c+1
...
ck+1
c(k-1)+1
...
c+1
1
...
n
...
n-1
c
...
n+2
2
...
1
0
N=c(k+1)
N=2n
...
 Tree depth is c, a constant number,
independent with platoon size;
 Stability margin can be bounded away
from zero
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Performance Analysis considering V2V communication

3. Stability Margin Improvement – Topology Selection
0
10
-1
0.25
10
-2
Stability Margin
Stability Margin
0.2
0.15
0.1
0.05
0
0
10
-3
10
-4
10
Ω(N)=N/2,Tree
Ω(N)=N/4,Tree
Ω(N)=N/2,Tree
Ω(N)=N/4,Tree
-5
10
10
50
20
40
30
20
40
Tree Depth Parameter:c
50
30
10
0
Local Communication Range:h
 It is the tree depth 𝑐 rather than local
communication range ℎ that dominates
the stability margin.
-6
10
10
depth:
depth:
depth:
depth:
c=2
c=4
c=N/2
c=3N/4
100
The number of vehicles:N
500
 Extending information flow to reduce
the tree depth is one major way to
guarantee a bounded stability margin.
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Performance Analysis considering V2V communication
3. Stability Margin Improvement – Asymmetric Control

Consider a homogeneous platoon under the BD topology with the asymmetric controller
architecture given by
𝑋 = 𝐼𝑁 ⊗ 𝐴 − ℒ𝐵𝐷 + 𝒫𝐵𝐷 𝜖 ⨂𝐵𝑘 𝑇 𝑋
(3.1) For any fixed&ϵ ∈ 0,1 , the stability margin is bounded away from zero and
independent of the platoon size&𝑁 (N can be any finite integer).
kfN
kbN-1
N
kfN-1
N-1
kbN-2
...
kf3
kb2
kf2
2
kb1
kf1
1
...

0
LV
Asymmetric control
The controller is called asymmetric, if
𝑓
&𝑘𝑖 = 1 + 𝜖 𝑘, 𝑘𝑖𝑏 = 1 − 𝜖 𝑘&&&&𝑖 = 1, ⋯ , 𝑁 − 1
𝑓
&𝑘𝑁 = 1 + 𝜖 𝑘,&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
where&𝜖 ∈ 0,1 is called the asymmetric degree. Note that if 𝜖 = 0, then it is reduced
to the symmetric case.
State Key Lab of Automotive Safety and Energy· Tsinghua Univ.
30
Performance Analysis considering V2V communication
3. Stability Margin Improvement – Asymmetric Control- Proof

Eigenvalue Analysis
𝑠𝐼 − 𝐴 − 𝜆𝑖 𝐵𝑘 𝑇
= 𝑠3 +
Im
𝜆𝑖 𝑘𝑎 +1 2
𝑠
𝜏
+
𝜆𝑖 𝑘𝑣
𝑠
𝜏
+
𝜆𝑖 𝑘𝑝
𝜏
.
Stability Margin
dmin
Stability Margin

dmin = Re smin
= O 𝜆min .
R
For asymmetric control
dmin = O 𝜆min = constant number, For any fixed&ϵ ∈ 0,1
Independent with size

Tradeoff between Convergence Speed and Transient Performance
 Benefit: bounded stability margin,  good for convergence speed
 Cost: overshooting phenomena in transient process.
State Key Lab of Automotive Safety and Energy· Tsinghua Univ.
31
Performance Analysis considering V2V communication

3. Stability Margin Improvement – Asymmetric Control
30
15
1
6
12
18
24
30
10
Spacing Error (m)
10
-1
-2
10
0
0
-10
-5
10
-10
-20
-30
0
50
-3
100
Time (s)
150
200
0
50
(a)
10
100
Time (s)
150
200
(b)
25
-5
10
1
6
12
18
24
30
30
20
10
100
The number of vehicles:N
500
 The stability margin of a platoon with
asymmetric controllers is indeed bounded
away from zero and independent of with
the platoon size.
10
0
-10
1
6
12
18
24
30
20
15
10
Spacing Error (m)
-4
10
40
ε=0
ε=0.2
ε=0.4
ε=0.6
Spacing Error (m)
Stability Margin
10
5
1
6
12
18
24
30
20
Spacing Error (m)
0
5
0
-5
-10
-20
-15
-30
-40
-20
-25
0
50
100
Time (s)
150
200
(c)
0
50
100
Time (s)
150
200
(d)
Space errors for homogeneous platoon under BD
topology with different asymmetric&degree&ϵ.&(a)&ϵ=0&
(symmetric);&(b)&ϵ=0.2;&(c)&ϵ=0.4;&(d)&ϵ=0.6
State Key Lab of Automotive Safety and Energy· Tsinghua Univ.
32
Performance Analysis considering V2V communication
4. Linear Stable Controller Design - solving a Riccati equation

Consider a homogeneous platoon with linear controllers given by
𝑋 = 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 𝑋

Stability Region
&𝑝>0
𝑘
&𝑘𝑣 > 𝑘𝑝 𝜏 min 𝜆𝑖 𝑘𝑎 + 1
&𝑘𝑎 > − 1 max 𝜆𝑖
How to choose a specific controller gain?

Controller design by solving a Riccati equation
𝐴𝑇 𝑃𝜀 + 𝑃𝜀 𝐴 − 𝑃𝜀 𝐵𝐵𝑇 𝑃𝜀 + 𝜀𝐼3 = 0,
Choosing the controller gain as

𝑘 𝑇 = 𝛼𝐵𝑇 𝑃𝜀
Influence of Information
flow topology
Closed-loop Stability requirements
𝛼≥
1
2 min 𝜆𝑖
𝑖∈𝒩
State Key Lab of Automotive Safety and Energy· Tsinghua Univ.
33
Performance Analysis considering V2V communication
4. Linear Stable Controller Design - Simulation
15
4.5
Spacing Error(m)
5
0
-5
1
3
5
7
10
4
3.5
3
Spacing Error(m)
1
3
5
7
10
10
1.2
0.8
2.5
2
1.5
1
-10
1
3
5
7
10
1
Spacing Error(m)

0.6
0.4
0.2
0.5
-15
0
20
40
60
Time (s)
100
-0.5
0
20
1
3
5
7
10
3
1
3
5
7
10
3
2.5
2
1.5
1
10
20
Time (s)
30
3
1
𝜀
0.5
0.5
22.5
0.5
0.5
0.5
1
1
1
1
1
1
1
3
5
7
10
3.5
2
40
𝛼
(c)
4
2.5
2
1.5
1
0
0.5
0.5
-1
0
-0.5
0
4.5
4
Spacing Error(m)
3.5
-0.2
60
(b)
5
4
40
Time (s)
(a)
4.5
Spacing Error(m)
80
Spacing Error(m)
-20
0
0
Topolo 𝜆𝑚𝑖𝑛
gy
PF
1
PLF
1
BD
0.022
BDL
1
TPF
1
TPLF
1
0
20
40
Time (s)
(d)
60
-2
0
0
20
40
Time (s)
(e)
60
-0.5
0
20
40
60
Time (s)
(f)
State Key Lab of Automotive Safety and Energy· Tsinghua Univ.
34
Outline
Background
Problem Statements
Performance Analysis considering V2V
communication
Conclusions
State Key Lab of Automotive Safety and Energy· Tsinghua Univ.
35
Conclusions
 Vehicular platoon can bring many potential benefits, e.g.,
 Improving traffic capacity; Enhancing highway safety; Reducing road congestion
 V2V communication can generate various types of topologies for platoon.
 For&vehicular&platoons&under&“homogeneity&+&linear&feedback”
 1) Stability Region Analysis
• Explicitly established the stabilizing thresholds of linear controller gains for
platoons under different information flow topologies.
 2) Stability Margin Scaling Trend
• Obtained stability margin scaling trend for platoons under two typical
topologies, i.e., Bidirectional Topology and Bidirectional-leader Topology.
 3) Stability Margin Improvement
• Proposed two basic ways to improve the stability margin, i.e., topology
selection and controller adjustment.
 4) Linear Stable Controller Design
• Converted the platoon control problem to a parametric algebraic Riccati
equation. The designed controllers can guarantee the internal stability for a
variety of topologies
State Key Lab of Automotive Safety and Energy· Tsinghua Univ.
36
Thanks for your attention
Q & A?
37
Appendix: Robust issue

Robust Performance

Disturbance with Finite Energy
+∞
𝑤𝑖 𝑡
1. First-to-last amplification factor (𝐴𝐹𝑓2𝑙 )
𝐴𝐹𝑓2𝑙

𝑝𝑁
= sup
𝑤1
ℒ2
ℒ2
= 𝐺𝑓2𝑙 𝑠
ℋ∞
ℒ2
=
0
𝑤𝑖 𝑡
2
𝑑𝑡 < ∞
2. All-to-all amplification factor(𝐴𝐹𝑎2𝑎 )
𝐴𝐹𝑎2𝑎 = sup
𝑝
𝑊
ℒ2
ℒ2
= 𝐺𝑎2𝑎 𝑠
ℋ∞
Dynamics in Frequency domain
𝑋 = 𝐼𝑁 ⨂𝐴 − ℒ + 𝒫 ⨂𝐵𝑘 𝑇 ∙ 𝑋 + 𝐵 ∙ 𝑊
Time domain
Frequency domain
𝑌 =𝐶∙𝑋
ℒ 𝑌
𝐺 𝑠 =
= 𝐶 𝑠𝐼3𝑁 − 𝐴𝑐𝑙 −1 𝐵
ℒ 𝑊
= 𝐼𝑁 ⊗ 𝜏𝑠 3 + 𝑠 2 + ℒ + 𝒫 ⊗ 𝑘𝑝 + 𝑘𝑣 𝑠 + 𝑘𝑎 𝑠 2
−1
 It’s&very&difficult&to&analysis&the&ℋ∞ norm of transfer function 𝐺 𝑠 under
general information flow topology.
State Key Lab of Automotive Safety and Energy· Tsinghua Univ.
38
Appendix: Robust issue

1. Scaling Trend of 𝓗∞ norms under PF topology
Consider a homogeneous platoon under the PF topology with linear controllers given
by
−1
𝐺 𝑠 = 𝐼𝑁 ⊗ 𝜏𝑠 3 + 𝑠 2 + ℒ + 𝒫 ⊗ 𝑘𝑝 + 𝑘𝑣 𝑠 + 𝑘𝑎 𝑠 2
(2.1) the amplification factors 𝐴𝐹𝑓2𝑙 and 𝐴𝐹𝑎2𝑎 satisfy the following conditions
𝛽2 𝛼 𝑁 − 1
𝑁−1
𝑁−1
𝑁−1
𝛽1 𝛼
≤ 𝐴𝐹𝑎2𝑎 ≤
𝛽1 𝛼
≤ 𝐴𝐹𝑓2𝑙 ≤ 𝛽2 𝛼
𝛼−1
where, 𝛽1 , 𝛽2 is constant real number, 𝛼&>1.
N
N-1
N-2
1
...
0
Fig. Predecessor-following (PF) topology
Remarks
 The amplification factors will exponentially grow with increasing platoon size.
 These results are independent with controller gains, which means this is a fundamental
drawback of PF topology when using identical linear controller
State Key Lab of Automotive Safety and Energy· Tsinghua Univ.
39
Appendix: Robust issue

1. Scaling Trend of 𝓗∞ norms under PF topology-simulation results
N-1
N
N-2
1
0
...
3
10
3
Bode Diagram
10
A f2l
80
N=22
N=18
N=14
N=10
N=6
N=2
Upper bound: β2*α N-1
60
Lowerbound: β1*α N-1
1
10
0
10
10
40
Magnitude (dB)
a2a
2
Amplification factor:A
Amplification factor： A
f2l
2
10
1
10
0
A a2a
10
20
0
-20
Upper bound: β3*(α N-1)
-1
10
Lower bound: β1*α N-1
-1
0
5
10
15
Platoon size: N
20
25
10
0
5
10
15
Platoon size: N
20
-40
25
-2
10
-1
10
0
10
1
10
Frequency (rad/s)
 The amplification factors&𝐴𝐹𝑓2𝑙 and 𝐴𝐹𝑎2𝑎 indeed exponentially grow with
increasing platoon size.
State Key Lab of Automotive Safety and Energy· Tsinghua Univ.
40
Appendix: Robust issue
2. Scaling Trend of 𝓗∞ norms under PLF topology-simulation results

N-1
N
N-2
1
0
...
0
Bode Diagram
1
10
10
0
A f2l
Upper bound: β2*α1N-1
-1
10
-50
-100
10
-3
10
-4
10
Magnitude (dB)
a2a
Lower bound: γ 1*α1
-2
Amplification factor:A
Amplification factor： A
f2l
N-1
0
10
-200
N=22
N=18
N=14
N=10
N=6
N=2
-250
-300
A a2a
-5
10
-6
10
-150
-1
0
5
10
15
Platoon size: N
20
25
10
-350
Upper bound: γ3*(α1N-1)
Lower bound: γ2
0
5
10
15
Platoon size: N
20
-400
-2
10
25
-1
10
0
10
1
10
Frequency (rad/s)
 If the leader broadcast its information to all the following vehicle, resulting in
predecessor-leader following topology, then amplification factor will become better
State Key Lab of Automotive Safety and Energy· Tsinghua Univ.
41
Appendix: Robust issue
3. Scaling Trend of 𝓗∞ norms under BD topology

Consider a homogeneous platoon under the BD topology with linear controllers given
by
−1
𝐺 𝑠 = 𝐼𝑁 ⊗ 𝜏𝑠 3 + 𝑠 2 + ℒ + 𝒫 ⊗ 𝑘𝑝 + 𝑘𝑣 𝑠 + 𝑘𝑎 𝑠 2
(4.1) the amplification factor 𝐴𝐹𝑎2𝑎 at least increase with the platoon size as 𝑂 𝑁 2 &
N-1
N
N-2
1
0
...
A a2a
40
N=22
N=18
N=14
N=10
N=6
N=2
12
2
10
1
10
10
20
Magnitude (dB)
Amplification factor： A
a2a
30
f2l
Lower bound: N2/(k p*π2)
3
10
Amplification factor:A
Bode Diagram
14
4
10
8
6
10
0
4
0
10
-1
10
-10
2
0
0
10
1
10
Platoon size: N
0
5
10
15
20
Platoon size: N
25
30
-20
-2
10
State Key Lab of Automotive Safety and Energy· Tsinghua Univ.
-1
10
0
10
1
10
Frequency (rad/s)
42