june 2011disc school: traffic control1 networked control systems: modelling and control of road...

June 2011 DISC School: traffic control 1

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


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• ...


modeling of networked control systems

• decomposition

• abstraction


Road traffic networks: example of network of components, with simple dynamics

for each component,but with complicated behavior due to interactions between

components


traffic models

• urban traffic versus freeway traffic?

• boundaries of network/boundaries between components

• local variables and global variables

• level of detail of model?


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


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 speed reference trajectories


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


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)


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


simulation

• need efficient simulation for – analyzing performance– model predictive control design– particle filtering (simulation based state

estimation to be studied later on)


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)


freeway traffic

components: cells/on-and off-ramps

macroscopic model describes average speed and density


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)


Compositional representation of network

cell or section i

11, ii vN ii vN , 11, ii vN

mkmz ,1

Sensor measurements

1 i - 1 1

… i i + 1 … Nm

mkmz ,

Link m - 1 Link m Link m + 1

wi-1 wi

QiQi-1

traffic source or exit gate of upstream link

sink or entrance gate of downstream link


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:

),(),(),(

txx

txq

t

tx



• 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

• simplest static model:

v(x,t) = f((x,t))

v

vfree

cong

jam



• leads to 1st order partial differential equation:

v= f()

vfree

cong jam

),(),()).,((),(

txtxtxftt

tx


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

http://en.wikipedia.org/wiki/File:Fundamental_Diagram.PNG


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


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


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)


effect of speed limitations

• reduce vfree

• implies reduce

max flow

• but increase cong

v= f()

vfree

cong jam

q= .f()

qmax

congjam


control via speed limitations

• requires careful coordination of local control actions

• in order to achieve good performance of overall system


coordination

= synchronization of traffic lights

case study: coordinated control

of urban traffic


urban traffic network

Intersection 1

Intersection 2

Intersection 4

Intersection 3

Uncontrolled intersection 5

component 1

Intersection 1 Intersection 2

Intersection 3

Intersection 4

component 2


• 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 study

component models for urban traffic models


sensors and actuators

• controllable events = red/green switching at intersection

• one control agent per signalized intersection

• observable events = passage time of vehiclesat some sensor locations along links

• current status of traffic lights


urban traffic model: controlled intersection

controlled, timed automaton describes state of traffic light

green NS,

red EW

red NS, green

EW,

yellowNS,

red EW

red NS,yellow

EW

controllabletransition

uncontrollable transition

intersection with only 2 phases

ordering of phases fixed, but cycle time not fixed ; some phases may be skipped


control decisions

• model allows at each intersection GYR switching at any time

• team of control agents minimizes delays subject to – constraints imposed by supervisor– safety constraints – coordination requirements imposed by neighboring

intersections


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...


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


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


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 vehicle2-roadside and vehicle2vehicle communication (get driver out of the loop, except for safety)

• benchmark for switching control?


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


local models

queue with time-varyingarrival stream, service rate ON/OFFcontrollable

Q(t) vehiclespoint processA(t) of arriving platoons

(t)outflow with rate (t) = (t) or (t)

ON/OFF:(t) = max or 0

Intersection 1

Intersection 2

Intersection 4

Intersection 3

Uncontrolled intersection 5


urban traffic: queueing at node

greenred +yellow

vehicle based model

greenred +yellow

platoon based model

a platoon of 4vehicles arrives during this interval of time

a platoon of 8 vehicles, incl. platoon with 2 new arrivals,departs during this interval

Q0 = 2


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) )


• 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

platoon based models


state of system = { complete characterization of all platoons in system

+ size of each queue

+ state of traffic light automata }



queueing dynamics: when head of platoon reaches tail of queue add size of arriving platoon

and

while green

subtract size of departing platoon



urban traffic: queueing at node

greenred +yellow

vehicle based model

greenred +yellow

platoon based model

a platoon of 4vehicles arrives during this interval of time

a platoon of 8 vehicles, incl. 2 separate arrivals,departs during this interval

Q0 = 2


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


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


simple case study for coordination

arrival streamEW

arrival stream SN

one-way traffic no loops!



arrival streamEW

arrival stream SN


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


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

urban traffic: arrival streamcoordination and anticipation


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)


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


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


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


controller interaction near saturation

controller at critical intersection selects GYR 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


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



arrival streamEW

arrival stream SN

critical intersection


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


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


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)


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


• proposed approach attempts to control behavior (arrival times) of vehicles without active roadside2vehicle

communication

• proposed approach may indirectly force vehicles to merge into platoons

design of coordinating and anticipating controller


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


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


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)


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


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


thanks for the patience!

questions?

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