controllability of large scale networks - math.unipd.itmotta/controllability-zampieri.pdf · sandro...
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Sandro Zampieri
Universita’ di Padova
In collaboration with
Fabio Paqualetti - University of California at Riverside
Francesco Bullo - University of California at Santa Barbara
Controllability of large scale networks
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Large scale networks
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US electric grid
References: controllability of complex networks
References: classical results
Problem formulation
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( + ) = ( ) + ( )
!
= [ · · · ]
=
!
""""""""""#
$
%%%%%%%%%%&
A controllability metric
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Controlled node
A controllability metric
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A controllability metric
! ( )
:=!!
=
=! ( )
! ( )
! ( )
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Conditions ensuring low controllability
( )
( )
< <
( ) := |{! ! !( ) : |!| " }|
! ( ) "( # )
( )
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US electric grid
Conditions ensuring low controllability
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Example: consensus with circle graph
=
!
""""""""""""#
/ / . . . . . . // / / · · · · · ·
/ / · · ·
· · · / // · · · · · · / /
$
%%%%%%%%%%%%&
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Example: consensus with circle graph
/
/
! /
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Example: consensus with circle graph
/
/
! /
!
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Example: consensus with circle graph
/
/
! /
!
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High controllability and controllers positioning
V = { , . . . , } V , . . . ,V
=
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"#· · ·
· · ·
$
%& , =
!
"#· · ·
· · ·
$
%& ,
( + ) = ( ) +'
!N( ) + ( ),
! { , . . . , } N := { : "= }
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High controllability and controllers positioning
! V"= ! V! , ! ! { , . . . , } "= !
B # V
B =!
=
B
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High controllability and controllers positioning
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High controllability and controllers positioning
( ) := ( )!!
!N( )
( + ) = ( ) + ( )
,
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High controllability and controllers positioning
! := (!! ( , ), . . . ,!! ( , )),
" :=
!
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" · · · "" · · · "
" "
$
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" = ! ( " )! ! !
! ( ) #!"! / !
,
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High controllability and controllers positioning
! :=
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"""#
! ! · · · ! !! ! · · · ! !
! ! ! ! · · ·
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!̄ = max{! ( ) : " { , . . . , }} <
! ( ) # ( $ !̄ )
!"!!!!!!,
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High controllability and controllers positioning
! "
"
! ( ) ! ( " !̄ )
#!#!#"#!,! ( ) !
"!" / ",
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Examples
2 4 6 8 10 12 14 16 18 2040
30
20
10
0
N 2 4 6 8 10 12 14 16 18 20
40
30
20
10
0
nb
number of subsystems dimension of subsystems
! ( )
! ( )
! ( )
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Examples
! ( )
! ( )
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 160
50
40
30
20
10
0
10
n
m
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Examples
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 160
50
40
30
20
10
0
10
n
! ( )
! ( )
m
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Examples
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 160
50
40
30
20
10
0
10
n
! ( )
! ( )
m
ConclusionsSimilar results for observability
For controllability we need to control a fixed fraction of nodes
The decoupled control strategy works well for graph that are partitionable
The decoupled control strategy admits a decoupled control synthesis
Random positioning works pretty well
Phase transition can be noticed (critical fraction of controlled nodes)
There are a lot of open problems:
Controllability of random and of structured graphs
Performance of random positioning
Phase transition
Different metrics
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Thank you
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