Networked control systems:
Modelling and Control
of road traffic
René Boel
with thanks to
N. Marinica, M. Moradzadeh, and H. Sutarto
SYSTeMS Research Group, UGent
June 2011
DISC School: traffic control
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examples of
networked control systems
• freeway traffic with on-ramp metering
• stabilization of frequency and voltage in power
transmission and distribution network
• irrigation networks
• communication networks
• autonomous vehicles jointly carrying out a
complicated task
• macro-economic models
• ...
June 2011
DISC School: traffic control
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modeling of networked control
systems
• decomposition
• abstraction
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DISC School: traffic control
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Road traffic networks:
example of network
of components,
with simple dynamics
for each component,
but with complicated behavior
due to interactions between
components
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DISC School: traffic control
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traffic models
• urban traffic versus freeway traffic?
• boundaries of network/boundaries
between components
• local variables and global variables
• level of detail of model?
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sensors and actuators
for urban traffic
• sensors:
– (current) video cameras, radars measuring queues
at intersections
– (current) magnetic loops counting number of
vehicles passing location per time unit
– (future) GPS/radio communication vehicle/roadside
• actuator:
– (current) traffic lights
– (future) reference trajectories communicated to
vehicles equiped with collision avoidance tools
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sensors and actuators
for freeway traffic
• sensors:
– (current) magnetic loops counting number of
vehicles passing location per time unit
– (future) vehicle-to-vehicle and vehicle-to-roadside
communication
• actuator:
– (current) adjustable speed limitations
– (current) on-ramp metering
– (current) varaible message signs (routing advice)
–
(future)
platooning
and
June
2011
DISC
School:speed
traffic controlreference trajectories
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traffic networks
• interaction with environment:
– inflow and outflow of vehicles at edges of network,
– road conditions (weather, accidents,...)
• state variables for different components
– microscopic model: (velocity position) for each
vehicle in each component
– macroscopic model: (flow, density, speed) at each
location along each link
– macroscopic model intersection: queue sizes
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DISC School: traffic control
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traffic networks
• components:
– links (connecting entrance and exit point, connected
to source, sink, or exit/entrance point of other link
– on and off ramps along freeways (sources/sinks),
with or without on-ramp metering
– controlled intersections (traffic lights)
– uncontrolled intersections (priority rules)
– round-abouts
– sources and sinks of traffic (edges of network)
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DISC School: traffic control
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traffic networks
• components interconnected via
entrance/exit nodes (ports in bond graph
terminology)
• interaction between components: traffic
flow from exit point of one component to
entrance point of downstream component
connected to it
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simulation
• need efficient simulation for
– analyzing performance
– model predictive control design
– particle filtering (simulation based state
estimation to be studied later on)
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abstraction
• keep number of variables small
– macroscopic models with speed/density per
link/cell
– platoon based models rather than vehicle
based models
– fluid flow models (fluid PN, Markov modulated
arrival streams)
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freeway traffic
components: cells/on-and off-ramps
macroscopic model describes average
speed and density
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freeway traffic
macroscopic model:
• characterize state of system by defining for
each cell
– number of vehicles Ni(t) in cell i at time t
(if length of cell is L, and cell i has pi lanes in
parallel, this defines density i(t) = Ni(t)/L.pi )
– average speed vi(t) of these Ni(t) vehicles
– flow of vehicles leaving cell qi(t) = i(t).vi(t)
(provided vehicles uniformly distributed over cell)
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Compositional representation of
network
Sensor measurements
Link m - 1
traffic source
or exit gate of
upstream link
Link m
…
1
i-1
i
i+1
…
sink or
entrance gate of
downstream link
Nm
1
zm 1, k m
z m, k m
cell or section i
Qi-1
Ni 1 , vi 1
Ni , vi
wi-1
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Link m + 1
Qi
Ni 1 , vi 1
wi
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classical partial differential
equation model
• cell size L  0
• state variables at location x at time t:
– density (x,t)  Ni(t)/Li
– speed v(x,t)
– flow q(x,t) = (x,t).v(x,t)
• flow model as in hydraulics: conservation
equation of fluid:
 ( x, t ) q ( x, t )

  ( x, t )
t
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DISC School: traffic control
x
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classical partial differential
equation model
• q(x,t) = v(x,t).(x,t)
• v(x,t) selected according to local traffic
conditions, and observed traffic conditions
ahead of position x
v
• simplest static model:
v(x,t) = f((x,t))
vfree

cong
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jam
classical partial differential
equation model
• leads to 1st order partial differential equation:
 ( x, t )

   f (  ( x, t )).  ( x, t )    ( x, t )
t
t
v= f()
vfree
cong
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jam
DISC School: traffic control

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fundamental
traffic diagrams
traffic behavior unstable:
same flow q(x,t)
can be realized by:
• v(x,t) large, (x,t) small
• v(x,t) small, (x,t) large
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feedback control
• flow rate always below maximal flow rate
qmax = vfree.cong
• try to control traffic flow so that always
(x,t)  cong
to avoid waste of capacity
• if for some location x, time  > t
simulator predicts (x,t) > cong(x), then
control action u(x,t): slow down traffic
upstream of x
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state feedback
• provided sufficiently accurate estimates of
current state are available
• model predictive control design possible
• take into account hard constraint
(x,t)  cong
• computationally hard: distributed control
needed
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DISC School: traffic control
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coordination control
• optimize traffic flow per region, but take
interaction between neignbouring regions
into account
• need "simple" model of traffic behavior in
order to allow fast simulation
• fast simulation allows fast prediction of
effects of different control actions
(variable routing signs, speed limitations,
on-ramp metering)
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effect of speed limitations
• reduce vfree
v= f()
vfree
cong
• implies reduce
q= .f()
max flow
qmax
• but increase cong
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DISC School: traffic control
jam
cong jam

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
control via speed limitations
• requires careful coordination of local
control actions
• in order to achieve good performance of
overall system
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case study:
coordinated control
of urban traffic
coordination
= synchronization of traffic lights
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urban traffic network
component 1
Uncontrolled
intersection 5
Intersection
1
Intersection
2
Intersection
4
Intersection
3
Intersection 1
Intersection 2
Intersection 3
Intersection 4
component 2
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component models
for urban traffic models
• intersections with traffic lights
= modeled as queues (integrators)
competing for server capacity
(ON/OFF allocation by traffic signal)
• linking roads
modelled by delay + noise
i(t) = in(i)(t-) + noise
(connects upstream queue in(i) to queue i)
• arrival streams of vehicles at entrance points
at border of network under
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study
sensors and actuators
• controllable events = red/green switching
at intersection
• one control agent per signalized intersection
• observable events = passage time of
vehicles
at some sensor
locations along links
• current status of traffic lights
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urban traffic model:
controlled intersection
controlled, timed automaton describes
state of traffic light
intersection with
only 2 phases
green
NS,
red EW
red NS,
yellow
EW
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yellow
NS,
red EW
red NS,
green
EW,
DISC School: traffic control
controllable
transition
uncontrollable
transition
ordering of phases fixed,
but cycle time not fixed ;
some phases may be skipped
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control decisions
• model allows at each intersection GYR
switching at any time
• team of control agents minimizes delays
subject to
– constraints imposed by supervisor
– safety constraints
– coordination requirements imposed by neighboring
intersections
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DISC School: traffic control
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formalization of distributed
supervisory control problems
• supervisor translates specifications for global
task into local specifications for local subtasks
• topic of this part of lecture: design of local
control agents,
• local agent selects next action to be taken by
local actuator, using only...
June 2011
DISC School: traffic control
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information available to local
control agent
• knowledge about local model and about
model of interaction with neighboring
nodes
• measurements obtained by local sensors
 local state estimators (using local model
based algorithms)
• messages from neighboring agents and
from supervisor
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DISC School: traffic control
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coordination = synchronization
• global task executed successfully if all
subtasks are successfully executed by
local agent
• actions of neighboring agents must be
synchronized so that agents
make it as easy as possible
for their neighbors to
satisfy subtask
specifications
June 2011
DISC School: traffic control
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future control paradigm for
urban traffic?
• better performance possible if size and speed
of platoons of vehicles adapted to traffic lights
(current paradigm: adapt traffic lights to driver's
decisions)
• control speed of vehicles, using vehicle2roadside and vehicle2vehicle communication
(get driver out of the loop, except for safety)
• benchmark for switching control?
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how distributed can controller be
in order to achieve
satisfactory coordination?
• how much control can/must be delegated by
supervisor to local controller?
• this lecture only about structure of local agents,
and their communication requirements for
coordination
 very few formulas, but emphasis on models
since coordination depends on anticipation
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DISC School: traffic control
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local models
Uncontrolled
intersection 5
Intersection
1
Intersection
2
Intersection
4
Intersection
3
point process
A(t) of arriving
platoons
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queue with time-varying
arrival stream,
service rate ON/OFF
controllable
Q(t) vehicles
ON/OFF:
DISC(t)
School:=
traffic
controlor 0
max
(t)
outflow
with rate
(t) = (t) or
(t)
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urban traffic: queueing at node
a platoon of 4
vehicles arrives
during this interval of time
Q0 = 2
green
red +
yellow
vehicle based model
June 2011
a platoon of 8 vehicles,
incl. platoon with 2 new arrivals,
departs during this interval
green
red +
yellow
platoon based model
DISC School: traffic control
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platoon based models
• platoon = group of vehicles travelling close
together at approximately same speed
• platooni at time t characterized by
= (ni(t) = number of vehicles in platoon i,
ℓi(t) = last sensor location passed by head
of platoon i,
i,head(t), i,tail(t) times at which head/tail of
platoon i passed (or will pass)
this location ℓi(t)
)
June 2011
DISC School: traffic control
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platoon based models
• event based models, with as events:
– arrival times of platoons at entrance points
– passage time of heads of platoon at sensor
locations, and at intersections
– G/Y/R switching time
• system noise generates random changes in
– platoon size traveling though links, or being
split up at intersections
– travel time of platoons through links
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DISC School: traffic control
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platoon based models
state of system = { complete characterization
of all platoons in system
+ size of each queue
+ state of traffic light
automata
}
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platoon based models
queueing dynamics:
when head of platoon reaches tail of queue
add size of arriving platoon
and
while green
subtract size of departing platoon
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urban traffic: queueing at node
a platoon of 4
vehicles arrives
during this interval of time
a platoon of 8 vehicles,
incl. 2 separate arrivals,
departs during this interval
Q0 = 2
green
red +
yellow
vehicle based model
June 2011
green
red +
yellow
platoon based model
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coordination and anticipation
• trajectory shown in previous example:
very badly synchronized traffic lights!
– starvation: traffic light green when queue empty and
no arriving platoon
– predictable, large platoons arrive at red light
 long queues at next R/Y/G switch
 extra delay due to acceleration of
vehicles
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DISC School: traffic control
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c-rule
myopic (suboptimal) strategy applies c-rule:
switch to that phase in cycle of traffic light
that on the average
provides
largest instantaneous reduction
of backlog of waiting vehicles
optimal only under unrealistic assumptions
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simple case study for
coordination
arrival
stream
EW
one-way traffic
no loops!
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DISC School: traffic control
arrival stream SN
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simple case study for
coordination
arrival
stream
EW
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DISC School: traffic control
arrival stream SN
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good feedback control actions
• feedback control strategies under investigation
should adapt switching times of traffic lights
to arrival times of platoons of vehicles
• need information about approaching platoons
• why deviate from c-rule?
– loss at yellow period  don't switch too often
– avoid acceleration delay
– avoid long queues blocking upstream intersections
June 2011
DISC School: traffic control
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urban traffic:and
arrival
stream
coordination
anticipation
intuitively model to be used in design and
coordination control strategy depends
strongly on load:
– light load only requires anticipation, no
coordination needed
– near saturation: both anticipation and
coordination needed, due to stochasticity
– saturated system need not take into account
stochastic phenomena, coordination important
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urban traffic: very light load
• switch traffic light to green only when platoon is
approaching
• probability of arriving platoon in conflicting
direction is negligible during this green period
• almost all platoons pass without any queueing
or acceleration delay
• no coordination among neighboring
intersections needed (not even a common cycle
time is necessary)
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what to do as traffic load
increases?
• rush hour builds up by congesting a few "critical
intersections"
• coordination between switching times of traffic
light at neighbouring intersections must
– avoid waste of capacity due to starvation at critical
intersections
– thus preventing spread of congestion as long as
possible
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supervisor
selects critical intersections
• delay along heavily loaded origin-destination
pairs of traffic flow will cause greatest risk for
spreading congestion
• critical intersections where two heavily
loaded OD-pairs cross each other,
must act as "master" control agent
imposing extra rules on
neighboring "slave" control agents
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supervisor
selects critical intersections
• supervisor acts on slower time scale than
local controllers
• supervisor uses average arrival rates per
OD pair
• average waiting time argument was based
on stationary regime, covering several
cycles of the traffic lights
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DISC School: traffic control
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controller interaction near
saturation
controller at critical intersection selects
GYR switching time so that
– red-green cycle within bounds imposed by
supervisor
– minimize waste of capacity due to starvation
model used in control design (e.g. for MPC)
must allow prediction of arrival times of
platoons at intersections
June 2011
DISC School: traffic control
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traffic model
for coordination control
for heavy load
• timed Petri net (or timed automaton)
model each intersection,
with coloured tokens
• token value = size of platoon
• random delay model for road
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simple case study for
coordination
critical
intersection
arrival
stream
EW
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DISC School: traffic control
arrival stream SN
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state based cordination control
• local feedback controller needs estimate of
expected arrival time and size of all
platoons approaching intersection
• to be estimated from on-line data obtained
from sensors, using recursive Bayesian
filter, e.g. particle filter
June 2011
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particle based optimization
• local coordinating control agent requires
information on estimated current local state
select switching time of traffic lights so that
– (supervisory design) specifications are met
– (optimal team theory) delays are minimized
for model based predictions for most likely
trajectories of platoons
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DISC School: traffic control
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design of coordinating and
anticipating controller
• controller at "master" intersection informs
neighbors of earliest/latest time [en,ln] they
should send next n-th platoon
so that master can meet all its specifications
without any waste of capacity
• taking into account travel times along links
• requires solution of large set of linear
inequalities (e.g. via distributed CSP)
June 2011
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artificial platoons and
abstraction
• modeling: reduce complexity of
combinatorial problem by grouping vehicles
into larger platoons that travel together as
"indivisible atoms"
• operational implementation: merge small
platoons together in one large platoon
behind red traffic lights at non-critical
intersections
June 2011
DISC School: traffic control
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design of coordinating and
anticipating controller
• proposed approach attempts to control
behavior (arrival times) of vehicles
without active roadside2vehicle
communication
• proposed approach may indirectly force
vehicles to merge into platoons
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DISC School: traffic control
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congested traffic
under very heavy load
average streams of traffic already
lead to congestion of network:
controllers must stabilize
deterministic system
described by fluid flow model
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DISC School: traffic control
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urban traffic: congested mode
• cycle time fixed to maximal allowable value
(pedestrians; blocking upstream intersections)
• local control decision: allocate green fraction for
each phase to reduce as much as possible the
total queue size, averaged per cycle,
• using as information the locally observed arrival
rates in each direction
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DISC School: traffic control
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urban traffic: congested mode
• supervisor decision:
impose common cycle length to intersections,
and locally optimal R/G split, using
information on inflow rates at edges of network
• communication between neighboring agents
only to avoid starvation, by selecting offset
between traffic lights at neighboring
intersections (adapt to travel time along link)
June 2011
DISC School: traffic control
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traffic model for coordinating
control for congested traffic
• fluid flow model, e.g. fluid Petri net
• analysis simplified: congestion means that
token load in many places always > 0,
i.e. many transitions always enabled
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conclusions
• progress in application of distributed
supervisory control by selecting simple
examples where
coordination
can significantly improve performance
• simple case study confirms philosophy of
• robustly optimizing local actions
• coordination through exchange of
specifications
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thanks for the patience!
questions?
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