Controllability of large scale networks

(1)

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

1

(2)

Large scale networks

2

US electric grid

(3)

References: controllability of complex networks

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ZSP RS T I

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[SVOW EV<MZTVITVMRXEV<MZ

(4)

References: classical results

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& 1EV\ ( /SIRMK ERH( +ISVKIW DD3TXMQEPWIRWSVERHEGXYEXSVPSGEXMSRJSVHIWGVMTXSVW]WXIQW YWMRKKIRIVEPM^IH+VEQMERWERHFEPERGIHVIEPM^EXMSRW MR %QIVMGER'SRXVSP'SRJIVIRGI &SWXSR 1%

97%.YP] TT

, 67LEOIVERH1 8ELEZSVM DD3TXMQEPWIRWSVERHEGXYEXSVPSGEXMSRJSVYRWXEFPIW]WXIQW .SYVREP SJ:MFVEXMSRERH'SRXVSP ?3RPMRIA %ZEMPEFPI

( +ISVKIW DD8LIYWISJSFWIVZEFMPMX]ERHGSRXVSPPEFMPMX]+VEQMERWSVJYRGXMSRWJSVSTXMQEPWIR

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3VPIERW 0%97%(IG TT

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/ & 0MQ DD1IXLSHJSVSTXMQEPEGXYEXSVERHWIRWSVTPEGIQIRXJSVPEVKI¾I\MFPIWXVYGXYVIW %-%%

.SYVREPSJ+YMHERGI 'SRXVSP ERH(]REQMGW ZSP RS TT 

(5)

Problem formulation

5

'SRWMHIVEPMRIEVW]WXIQ

\(X + ) = %\(X) + &Y(X)

[LIVI % MWWTEVWI R×R QEXVM\XLIMRXIVEGXMSRWFIX[IIRXLIWXEXIWMWHIWGVMFIH F]EKVETL ERH

& = [I

_M

· · · I

^M^Q

] [LIVI

I

_M

=



 





 



M

(6)

A controllability metric

6

-JXLIKVETLMWWXVSRKP]GSRRIGXIHERHXLIVIXLIWIPJPSSTWXLIRXSLEZIXLEX XLIVIWYPXMRKW]WXIQMWGSRXVSPPEFPIKIRIVMGEPP]MRXLIRSR^IVSIRXVMIWSJ % MX MWIRSYKLXLEXSRP]SRIWXEXIMWGSRXVSPPIH

Controlled node

(7)

A controllability metric

7

59)78-32 ,S[GSRXVSPPEFPIMWXLIVIWYPXMRKW]WXIQ#

(8)

8

A controllability metric

8LIVIEVIZEVMSYWGLSMGIWSJQIXVMGWHIWGVMFMRKLS[GSRXVSPPEFPIEW]WXIQMW

;IGLSSWIXLIQMRMQYQIMKIRZEPYISJXLIGSRXVSPPEFMPMX]+VEQMER λ

_QMR

(;

₈

) [LIVI

;

₈

:=

!

8−

X=

%

^X

&&

⁸

%

^X

[IEVIEWWYQMRKXLEX % MWW]QQIXVMG

8LIIRIVK]XSHVMZIXLIWXEXIJVSQ^IVSXSERSVQSRIWXEXIMRXLI[SVWX GEWI MWKMZIRF]

) =

λ

_QMR

(;

₈

)

LMKL λ

_QMR

(;

₈

)

PS[ λ

_QMR

(;

₈

) PS[GSRXVSPPEFMPMX]

LMKLGSRXVSPPEFMPMX]

(9)

9

Conditions ensuring low controllability

'SRWIUYIRGIW 7MRGI R(') [LMGLX]TMGEPP]KVS[PMRIEVP]MR R XLIR

JSV½\IH Q XLIHIKVIISJGSRXVSPPEFMPMX]HIGVIEWIWEXPIEWXI\TSRIRXMEPP]

MR R(') 8LIVIJSVI X]TMGEPP] JSV ½\IH Q XLI HIKVII SJ GSRXVSPPEFMPMX]

HIGVIEWIWEXPIEWXI\TSRIRXMEPP]MR R

-RSVHIVXSLEZIE½\IHHIKVIISJGSRXVSPPEFMPMX][IRIIHXSGSRXVSPE

½\IHJVEGXMSRSJWXEXIW

*M\ER]GSRWXERX < ' < ERHPIX

R(') := |{λ ∈ λ(%) : |λ| ≤ '}|

8LIR

λ

_QMR

(;

₈

) ≤

'

( − '

) '

^R(')^Q

(10)

10

US electric grid

Conditions ensuring low controllability

(11)

11

Example: consensus with circle graph

% =



 



/ / . . . . . . /

/ / / · · · ·

/ / ···

· · · / /

/ · · · / /



 



(12)

12

Example: consensus with circle graph

/

−/

)MKIRZEPYIWSJ %

(13)

13

Example: consensus with circle graph

/

−/

O

λ

_O

)MKIRZEPYIWSJ %

(14)

14

Example: consensus with circle graph

/

−/

'

−'

)MKIRZEPYIWSJ %

(15)

15

High controllability and controllers positioning

(IGSYTPIHGSRXVSPWXVEXIK]

2IX[SVOTEVXMXMSRMRK 4EVXMXMSR V = {, . . . , R} MRXS 2 HMWNSMRXWIXW V

, . . . , V

²

%JXIVVIPEFIPMRKSJWXEXIWERHMRTYXW XLIQEXVMGIWVIEHEW

% =



 

%

· · · %

²

%

₂

· · · %

₂



  , & =



 

&

· · ·

· · · &

²



  ,

8LIRIX[SVOWH]REQMGWGERFI[VMXXIREWXLIMRXIVGSRRIGXMSRSJ 2 WYFW]WXIQW SJXLIJSVQ

\

_M

(X + ) = %

_M

\

_M

(X) + '

N∈N^M

%

_MN

\

_N

(X) + &

_M

Y

_M

(X),

[LIVI M ∈ {, . . . , 2} ERH N

^M

:= {N : %

^MN

"= }

(16)

16

High controllability and controllers positioning

(IGSYTPIHGSRXVSPWXVEXIK]

7IPIGXMSRSJXLIGSRXVSPRSHIW ;IWE]XLEXERSHI M ∈ V

^O

MWE FSYRHEV]

RSHIMJ E

_MN

"= JSVWSQIRSHI N ∈ V

^!

[MXL O, ! ∈ {, . . . , 2} ERH O "= !

0IX B

^M

⊆ V

^M

FIXLIWIXSJFSYRHEV]RSHIWMRXLI MXLGPYWXIV ERHPIX

B =

!

2 M=

B

^M

FIXLIWIXSJEPPXLIFSYRHEV]RSHIWSJXLITEVXMXMSR ;IWIPIGXXLIWIXSJ

GSRXVSPRSHIWXSFIEWIXGSRXEMRMRKXLIFSYRHEV]RSHIW

(17)

17

High controllability and controllers

positioning

(18)

18

High controllability and controllers positioning

(IGSYTPIHGSRXVSPWXVEXIK]

**8LIHIGSYTPIHGSRXVSPPE[ *SVXLITVIZMSYWTEVXMXMSRIHW]WXIQGSRWMHIV** XLIMRTYXW

Y

_M

(X) := Z

_M

(X) − !

N∈N^M

&

⁸_M

%

_MN

\

_N

(X)

2SXMGIXLEXXLMWGSRXVSPPE[]MIPHWERI[W]WXIQGSQTSWIHF] 2 HIGSYTPIH WYFW]WXIQW

\

_M

(X + ) = %

_M

\

_M

(X) + &

_M

Z

_M

(X)

8LI½REPWXITMWXSGLSSWI Z

_M

[LMGLQMRMQM^IWXLIIRIVK]XSWXIIVXLIWYFW]W

XIQXSXLIHIWMVIHWYFWXEXI 8LMW[MPPHITIRHSRXLIGSRXVSPPEFMPMX]+VEQMER

;

_M,8

XLIXLI MXLWYFW]WXIQ

(19)

19

High controllability and controllers positioning

(I½RI

Λ := HMEK(λ

⁻_QMR

(;

_,8

), . . . , λ

⁻_QMR

(;

_2,8

)),

Γ :=



 



γ

· · · γ

²

γ

· · · γ

²

γ

₂

γ

₂



 

 ,

[LIVI

γ

_MN

= !&

⁸M

%

_MN

(^- − %

^N

)

⁻

&

_N

!

^,^∞

8LISVIQ -J[IGLSSWIEHIGSYTPIHGSRXVSPPE[XLIR[ISFXEMR λ

_QMR

(;

₈

) ≥

!ΓΛ

^/

!

,

(20)

20

High controllability and controllers positioning

(I½RI

∆ :=



 

 

!%

!

· · · !%

²

!

!%

!

· · · !%

²

!

!%

²

!

!%

²

!

· · ·



 

  .

ERHEWWYQIXLEX

λ ¯

_QE\

= max {λ

^QE\

(%

_M

) : M ∈ {, . . . , 2}} <

8LISVIQ -J[IGLSSWIEHIGSYTPIHGSRXVSPPE[XLIR[ISFXEMR

λ

_QMR

(;

₈

) ≥ ( − ¯λ

^QE\

)

!Λ!

∞

!∆!

_∞

,

(21)

21

High controllability and controllers positioning

'SRWIUYIRGIW -RSVHIVXSLEZILMKLGSRXVSPPEFMPMX]MXMWGSRZIRMIRXXSTS

WMXMSRXLIGSRXVSPPIVWMRWYGLE[E]XLEX

8LI QEXVMGIW Λ SV ∆ EVIWQEPP [LMGLQIERWGLSSWMRKWYFW]WXIQW[LMGL EVI[IEOP]MRXIVGSRRIGXIH

8LIQEXVM\ ∆ MWWQEPP [LMGLQIERWGLSSWMRKGSRXVSPPIVWMRXLIWYFW]WXIQW QEOMRKXLIQLMKLP]GSRXVSPPEFPI

λ

_QMR

(;

₈

) ≥ ( − ¯λ

^QE\

)

#Λ#

∞

#∆#

_∞

, λ

_QMR

(;

₈

) ≥

"ΓΛ

^/

"

,

(22)

22

Examples

'MVGYPERXKVETL

2 4 6 8 10 12 14 16 18 20

ï40 ï30 ï20 ï10 0

N ₂ ₄ ₆ ₈ ₁₀ ₁₂ ₁₄ ₁₆ ₁₈ ₂₀

ï40 ï30 ï20 ï10 0

n_b

number of subsystems dimension of subsystems

λ

_QMR

(;

₈

) [MXLXLIHIGSYTPIHGSRXVSPWXVEXIK]

PS[IVFSYRHSJ λ

_QMR

(;

₈

) [MXLXLIHIGSYTPIHGSRXVSPWXVEXIK]

λ

_QMR

(;

₈

) [MXLVERHSQTSWMXMSRMRK

(23)

23

Examples

λ

_QMR

(;

₈

) [MXLXLIHIGSYTPIHGSRXVSPWXVEXIK]

λ

_QMR

(;

₈

) [MXLVERHSQTSWMXMSRMRK

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ï60 ï50 ï40 ï30 ï20 ï10 0 10

n

4S[IVKVMH[MXLRSHIW

m

(24)

24

Examples

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ï60 ï50 ï40 ï30 ï20 ï10 0 10

n

)TMHIQMGWRIX[SVO[MXLRSHIW

λ

_QMR

(;

₈

) [MXLXLIHIGSYTPIHGSRXVSPWXVEXIK]

λ

_QMR

(;

₈

) [MXLVERHSQTSWMXMSRMRK

m

(25)

25

Examples

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

ï60 ï50 ï40 ï30 ï20 ï10 0 10

n

7SGMEPRIX[SVO[MXLRSHIW

λ

_QMR

(;

₈

) [MXLXLIHIGSYTPIHGSRXVSPWXVEXIK]

λ

_QMR

(;

₈

) [MXLVERHSQTSWMXMSRMRK

m

(26)

Conclusions

Similar 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

26

(27)

Thank you

27

Controllability of large scale networks

Sandro Zampieri

Universita’ di Padova

In collaboration with

Fabio Paqualetti - University of California at Riverside

Francesco Bullo - University of California at Santa Barbara