feedback control of scalar conservation ...nlimperis/iasson.pdffeedback control of scalar...

FEEDBACK CONTROL OF SCALAR

CONSERVATION LAWS WITH APPLICATION TO

DENSITY CONTROL IN FREEWAYS BY MEANS OF

VARIABLE SPEED LIMITS

Iasson Karafyllis and Markos Papageorgiou

SSppeeeedd lliimmiittss mmaayy vvaarryy……

SSppeeeedd lliimmiittss mmaayy bbee vveerryy ssmmaallll……

HHooww ccaann wwee sseett aapppprroopprriiaattee ssppeeeedd lliimmiittss??

OOuuttlliinnee

The model

OOuuttlliinnee

The model

Two Control Problems

OOuuttlliinnee

The model


Two Possible Solutions

OOuuttlliinnee

The model



Example

OOuuttlliinnee

The model



Example

Future Research

TThhee mmooddeell

( , ) ( , ) 0q

t x t xt x

the conservation law

TThhee mmooddeell

( , ) ( , ) 0q

t x t xt x


[0, ]x L , 0t

TThhee mmooddeell

( , ) ( , ) 0q

t x t xt x


[0, ]x L , 0t

max( , ) 0,t x , ( , ) ( , ) ( , )q t x t x v t x

TThhee mmooddeell

( , ) ( , ) 0q

t x t xt x


[0, ]x L , 0t

max( , ) 0,t x , ( , ) ( , ) ( , )q t x t x v t x

In the absence of speed limits

( , ) ( ( , ))q t x f t x

TThhee mmooddeell

( , ) ( , ) 0q

t x t xt x


[0, ]x L , 0t

max( , ) 0,t x , ( , ) ( , ) ( , )q t x t x v t x

In the absence of speed limits

( , ) ( ( , ))q t x f t x

(the standard LWR model)

TThhee mmooddeell When speed limits are present

( , ) ( ( , ), ( , ))q t x F t x l t x


( , ) ( ( , ), ( , ))q t x F t x l t x

speed limit at ( , )( , )

maximum legal speed

t xl t x

the speed limit ratio at ( , )t x


( , ) ( ( , ), ( , ))q t x F t x l t x


maximum legal speed

t xl t x


Assumption: ( , ) (0,1]l t x


( , ) ( ( , ), ( , ))q t x F t x l t x


maximum legal speed

t xl t x


Assumption: ( , ) (0,1]l t x

(the “mean” driver respects the law)

TThhee mmooddeell

We usually think that speed limits have to be piecewise constant both wrt space and time

TThhee mmooddeell


But speed limits don’ t have to be piecewise constant

TThhee mmooddeell



This is the era of automated and connected vehicles!

TThhee mmooddeell



This is the era of automated and connected vehicles!

The traffic control centre may communicate different variable speed limits to individual equipped vehicles at virtually arbitrary space, time and value resolution

TThhee mmooddeell Example: In Carlson, R. C., Mainstream Traffic Flow Control on Motorways, Ph.D. Thesis, Technical University of Crete, Chania, Greece, 2011

1( , ) exp

1

bF l A l

a al

, , 0A b , 0a


1( , ) exp

1

bF l A l

a al

, , 0A b , 0a

Define:

( ) : ( ,1)f F , for max[0, ]


1( , ) exp

1

bF l A l

a al

, , 0A b , 0a

Define:

( ) : ( ,1)f F , for max[0, ]

(the flow-density curve in absence of speed limits)

TThhee mmooddeell In general:

(H) The function 2max max[0, ];[0, ]f C q , where max 0q is a function for

which there exists max(0, )cr with the following properties:




(i) ( ) 0f for all [0, )cr , ( ) 0crf ,




(i) ( ) 0f for all [0, )cr , ( ) 0crf ,

(ii) ( ) 0f for all max[0, ] , and




(i) ( ) 0f for all [0, )cr , ( ) 0crf ,


(iii) (0) 0f and ( ) 0f for all max(0, ] .




(i) ( ) 0f for all [0, )cr , ( ) 0crf ,


(iii) (0) 0f and ( ) 0f for all max(0, ] .

(standard assumption for the fundamental diagram)

TThhee mmooddeell A major simplification:

For every (0,1]u , max0, there exists (0,1]l such that

( ) ( , )u f F l



( ) ( , )u f F l

We get:

( , ) ( , ) ( ( , )) 0t x u t x f t xt x



( ) ( , )u f F l

We get:

( , ) ( , ) ( ( , )) 0t x u t x f t xt x

[ ]t is the state (( [ ])( ) ( , )t x t x for [0, ]x L )



( ) ( , )u f F l

We get:

( , ) ( , ) ( ( , )) 0t x u t x f t xt x


[ ]u t is the distributed control input



( ) ( , )u f F l

We get:

( , ) ( , ) ( ( , )) 0t x u t x f t xt x


[ ]u t is the distributed control input

( , ) 0,1u t x

TTwwoo CCoonnttrrooll PPrroobblleemmss

max(0, ) : the set point.



1st Problem: Free Speed Limit at Inlet




Construct a feedback law

( , ) ( [ ], )u t x K t x , for ( , ) [0, ]t x L

with

( [ ], ) (0,1]K t x , for ( , ) [0, ]t x L




Construct a feedback law

( , ) ( [ ], )u t x K t x , for ( , ) [0, ]t x L

with

( [ ], ) (0,1]K t x , for ( , ) [0, ]t x L

so that

lim ( , )t

t x

, lim ( , ) 1

tu t x

, for [0, ]x L .


lim ( , ) 1t

u t x


lim ( , ) 1t

u t x

the freeway will practically operate without speed limits

after an initial transient period, i.e. after the problem has been tackled.

TTwwoo CCoonnttrrooll PPrroobblleemmss Inlet Flow: ( ,0) ( ( ,0))u t f t


may become small for a transient period


may become small for a transient period

queues may be created at the entrance of the freeway


2nd Problem: No Speed Limit at Inlet


2nd Problem: No Speed Limit at Inlet We additionally require:

( ,0) 1u t , for 0t (input constraint)


2nd Problem: No Speed Limit at Inlet We additionally require:

( ,0) 1u t , for 0t (input constraint)

( ,0)t , for 0t (boundary condition)

TTwwoo CCoonnttrrooll PPrroobblleemmss 1st Control Problem: Free of boundary conditions

TTwwoo CCoonnttrrooll PPrroobblleemmss 1st Control Problem: Free of boundary conditions 2nd Control Problem: One boundary condition at the entrance

TTwwoo CCoonnttrrooll PPrroobblleemmss 1st Control Problem: Free of boundary conditions 2nd Control Problem: One boundary condition at the entrance More demanding than the 1st problem


Inlet Flow: ( ,0) ( ( ,0)) ( )u t f t f


Inlet Flow: ( ,0) ( ( ,0)) ( )u t f t f

Both Control Problems are Nonlinear

TTwwoo PPoossssiibbllee SSoolluuttiioonnss Theorem (Global Asymptotic Stabilization of an arbitrary spatially uniform density profile in the sup norm for the 1st problem):

TTwwoo PPoossssiibbllee SSoolluuttiioonnss Theorem (Global Asymptotic Stabilization of an arbitrary spatially uniform density profile in the sup norm for the 1st problem): Suppose that Assumption

(H) holds. Let max(0, ) , 0,1/ ( )k L . Define

1

0

, 1 ( )

x

M w x k w s ds



1

0

, 1 ( )

x

M w x k w s ds

Then there exists max max: (0, ] (0, ]c q such that for every

10 max[0, ];(0, ]C L



1

0

, 1 ( )

x

M w x k w s ds

Then there exists max max: (0, ] (0, ]c q such that for every

10 max[0, ];(0, ]C L ((aallll pphhyyssiiccaallllyy rreelleevvaanntt pprrooffiilleess!!))

TTwwoo PPoossssiibbllee SSoolluuttiioonnss the initial value problem with 0[0] and

[0, ]

min ( ( , )) [ ],

( , )( ( , )) [ ],

z Lf t z M t z

u t xf t x M t x

, for 0t , [0, ]x L


[0, ]

min ( ( , )) [ ],

( , )( ( , )) [ ],

z Lf t z M t z

u t xf t x M t x

, for 0t , [0, ]x L

((tthhee ffeeeeddbbaacckk llaaww))


[0, ]

min ( ( , )) [ ],

( , )( ( , )) [ ],

z Lf t z M t z

u t xf t x M t x

, for 0t , [0, ]x L


has a unique solution 1max[0, ];(0, ]C L , which satisfies:

0[ ] exp( )t ct

, for all 0t


[0, ]

min ( ( , )) [ ],

( , )( ( , )) [ ],

z Lf t z M t z

u t xf t x M t x

, for 0t , [0, ]x L


has a unique solution 1max[0, ];(0, ]C L , which satisfies:

0[ ] exp( )t ct

, for all 0t

lim ( , ) 1t

u t x

, for all [0, ]x L

where 0[0, ]

min ( )z L

c c z

.

TTwwoo PPoossssiibbllee SSoolluuttiioonnss Theorem (Regional Exponential Stabilization of an arbitrary spatially uniform density profile in the sup norm for the 2nd problem):

TTwwoo PPoossssiibbllee SSoolluuttiioonnss Theorem (Regional Exponential Stabilization of an arbitrary spatially uniform density profile in the sup norm for the 2nd problem): Suppose that Assumption (H) holds.

Let max0,min , / 2cr .


Let max0,min , / 2cr . ((nnoott aann aarrbbiittrraarryy sseett ppooiinntt!!))



Then there exist constants , , 0 such that for every

10 max[0, ];(0, ]C L with 0(0) and

2

0 0 0

0

( ) ( ( ))2

xx

f s ds f x

, for [0, ]x L



Then there exist constants , , 0 such that for every

10 max[0, ];(0, ]C L with 0(0) and

2

0 0 0

0

( ) ( ( ))2

xx

f s ds f x

, for [0, ]x L

((nnoott aallll pphhyyssiiccaallllyy rreelleevvaanntt iinniittiiaall ccoonnddiittiioonnss!!))

TTwwoo PPoossssiibbllee SSoolluuttiioonnss

the initial-boundary value problem with 0[0] and

2

1

0

( , ) ( ( , )) ( , ) [ ]2

xx

u t x f t x f t s ds t

, for 0t , [0, ]x L



2

1

0

( , ) ( ( , )) ( , ) [ ]2

xx


, for 0t , [0, ]x L




2

1

0

( , ) ( ( , )) ( , ) [ ]2

xx


, for 0t , [0, ]x L


has a unique solution 1max[0, ];(0, ]C L , which satisfies

0 ( , ) 1u t x , for 0t , [0, ]x L ,



2

1

0

( , ) ( ( , )) ( , ) [ ]2

xx


, for 0t , [0, ]x L



0 ( , ) 1u t x , for 0t , [0, ]x L ,

0[ ] exp( )t t

, for all 0t ,



2

1

0

( , ) ( ( , )) ( , ) [ ]2

xx


, for 0t , [0, ]x L



0 ( , ) 1u t x , for 0t , [0, ]x L ,

0[ ] exp( )t t

, for all 0t ,

lim ( , ) 1t

u t x

, for all [0, ]x L .


How are shocks avoided for a scalar conservation law?



Let’s look at the closed-loop systems




1st problem: [0, ]

( , ) ( , ) min ( ( , )) ( [ ], )z L

t x k t x f t z M t zt

a zero-speed, first-order, hyperbolic PDE (with no bc’s applicable)




1st problem: [0, ]

( , ) ( , ) min ( ( , )) ( [ ], )z L



2nd problem: ( , ) ( , ) [ ]t x t x x tt

with ( ,0)t

a zero-speed, first-order, hyperbolic PDE (with one bc at 0x )




1st problem: [0, ]

( , ) ( , ) min ( ( , )) ( [ ], )z L




with ( ,0)t


TThhee ffeeeeddbbaacckk llaaww cchhaannggeess tthhee cchhaarraacctteerr ooff tthhee PPDDEE!!




1st problem: [0, ]

( , ) ( , ) min ( ( , )) ( [ ], )z L




with ( ,0)t


TThhee ffeeeeddbbaacckk llaaww cchhaannggeess tthhee cchhaarraacctteerr ooff tthhee PPDDEE!!

TThhee cclloosseedd--lloooopp ssyysstteemm iiss nnoo lloonnggeerr aa ccoonnsseerrvvaattiioonn llaaww!!


The state space for the 2nd problem:

2

1, max

0

: [0, ];(0, ] : (0) , ( ) ( ( ))2

xx

X C L f s ds f x



2

1, max

0

: [0, ];(0, ] : (0) , ( ) ( ( ))2

xx

X C L f s ds f x

it depends on the control parameters



2

1, max

0

: [0, ];(0, ] : (0) , ( ) ( ( ))2

xx

X C L f s ds f x


if 0 ,X then ,[ ]t X for all 0t



2

1, max

0

: [0, ];(0, ] : (0) , ( ) ( ( ))2

xx

X C L f s ds f x


if 0 ,X then ,[ ]t X for all 0t

we cannot handle every density profile simply by VSL (ramp metering may also be required)

EExxaammppllee

( ) expf with max 1.6 , 1L , 0.7

EExxaammppllee

( ) expf with max 1.6 , 1L , 0.7

For this case 1cr

EExxaammppllee

( ) expf with max 1.6 , 1L , 0.7

For this case 1cr

Controller parameters: 0.3k , 0.12 , 0.1

EExxaammppllee

( ) expf with max 1.6 , 1L , 0.7

For this case 1cr

Controller parameters: 0.3k , 0.12 , 0.1

Initial condition: 2 2

0( ) 4 (1.2 )x x x , [0, ]x L

EExxaammppllee

Feedback for the 1st control problem

Density Speed limit ratio

EExxaammppllee

Feedback for the 2nd control problem

Density Speed limit ratio

EExxaammppllee

Evolution of the sup-norm of the deviation

1st Control Problem 2nd Control Problem

EExxaammppllee

What if 1cr ? Only 1st control problem…

Density profiles

EExxaammppllee

Convergence faster for the 1st feedback (at the cost of possible queues)

EExxaammppllee


No need to use very small speed limits: speed limit ratios greater than 90% are capable of handling congestion phenomena

EExxaammppllee


No need to use very small speed limits: speed limit ratios greater than 90% are capable of handling congestion phenomena

Intuitive solutions: speed limit ratios smaller upstream the congestion

and higher downstream the congestion

FFuuttuurree RReesseeaarrcchh

Study the inhomogeneous case, i.e. consideration of a freeway stretch with spatially inhomogeneous flow-density relationships or freeways with multiple on-ramps and off-ramps



study of the effect of discretization




density profiles with discontinuities





output feedback stabilization problem





output feedback stabilization problem

VSL to 2nd order models?

THANK YOU!

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