1

USING SPATIAL SYMMETRIES TO USING SPATIAL SYMMETRIES TO DO RESOURCE ALLOCATIONDO RESOURCE ALLOCATIONIN LARGE SCALE SYSTEMSIN LARGE SCALE SYSTEMS

2

OBJECTIVESOBJECTIVES

SOLVE THE RESOURCE ALLOCATION PROBLEM FOR LARGE SCALE LINEAR SYSTEMS.(CONTROL WITH FLOW CONSTRAINTS)

SOLVE THE RESOURCE ALLOCATION PROBLEM SOLVE THE RESOURCE ALLOCATION PROBLEM FOR LARGE SCALE LINEAR SYSTEMS.FOR LARGE SCALE LINEAR SYSTEMS.(CONTROL WITH FLOW CONSTRAINTS)(CONTROL WITH FLOW CONSTRAINTS)

USE SYSTEM SPATIAL SYMMETRIES TO “BREAK”CONTROLLER DESIGN AND IMPLEMENTATIONINTO SMALLER PROBLEMS

USE SYSTEM SPATIAL SYMMETRIES TO USE SYSTEM SPATIAL SYMMETRIES TO ““BREAKBREAK””CONTROLLER DESIGN AND IMPLEMENTATIONCONTROLLER DESIGN AND IMPLEMENTATIONINTO SMALLER PROBLEMSINTO SMALLER PROBLEMS

USE SYSTEM SPATIAL SYMMETRIES AGAIN TOISOLATE FLOW CONSTRAINTSUSE SYSTEM SPATIAL SYMMETRIES AGAIN TOUSE SYSTEM SPATIAL SYMMETRIES AGAIN TOISOLATE FLOW CONSTRAINTSISOLATE FLOW CONSTRAINTS

3

MOTIVATIONMOTIVATION

LARGE SCALESYSTEM

LARGE SCALESYSTEM

SPATIAL SYMMETRIESSPATIAL SYMMETRIESSPATIAL SYMMETRIES

SMALL SCALESMALL SCALESYSTEMSYSTEM

SMALL SCALESMALL SCALESYSTEMSYSTEM

SMALL SCALESMALL SCALESYSTEMSYSTEM

SMALL SCALESMALL SCALESYSTEMSYSTEM

DECOUPLING INTODECOUPLING INTOSMALLER SYSTEMSSMALLER SYSTEMS

4

EXAMPLES OF SYSTEMS WITH SYMMETRYEXAMPLES OF SYSTEMS WITH SYMMETRY

FF EESECTORSECTOR

FF EESECTORSECTOR

FF EESECTORSECTOR

FF EESECTORSECTOR

FF EESECTORSECTOR

RESOURCES

RESOURCES

RESOURCES

RESOURCES

RESOURCES

RESOURCES

RESOURCES

RESOURCES

RESRES RESRES

DYNAMICS ISDYNAMICS ISINVARIANT UNDERINVARIANT UNDER909000 ROTATIONROTATION

5

EXAMPLES OF SYSTEMS WITH SYMMETRYEXAMPLES OF SYSTEMS WITH SYMMETRY

FF EESECTORSECTOR

FF EESECTORSECTOR

RESRES RESRES

INVARIANT UNDER REFLECTIONINVARIANT UNDER REFLECTIONAND PERMUTATIONSAND PERMUTATIONS

6

EACH SMALL SCALE SYSTEM GOVERNS THE DYNAMICS OF THE EACH SMALL SCALE SYSTEM GOVERNS THE DYNAMICS OF THE LARGE SCALE SYSTEM OVER A SPATIAL EIGENVECTOR.LARGE SCALE SYSTEM OVER A SPATIAL EIGENVECTOR.

EXAMPLES OF SPATIAL EIGENVECTORS ARE: EXAMPLES OF SPATIAL EIGENVECTORS ARE:

1 1 1 … 1 1 1 PHYSICAL MEANING:SPATIAL MEAN VALUE OVERALL SECTORS.

1 -1 0 .. 0 0 0 PHYSICAL MEANING:SHIFT OF RESOURCESFROM SECTOR 1 TO 2.

THE COMBINED EFFECT OF ALL SMALL SCALE SYSTEMS DESCRIBESTHE COMBINED EFFECT OF ALL SMALL SCALE SYSTEMS DESCRIBESTHE DYNAMICS OF THE OVERALL SYSTEM.THE DYNAMICS OF THE OVERALL SYSTEM.

EXAMPLES OF SYSTEMS WITH SYMMETRYEXAMPLES OF SYSTEMS WITH SYMMETRY

7

ISOLATING FLOW CONSTRAINTSISOLATING FLOW CONSTRAINTS

SMALL SCALESMALL SCALESYSTEMSYSTEM

SMALL SCALESMALL SCALESYSTEMSYSTEM

SMALL SCALESMALL SCALESYSTEMSYSTEM

SPATIALSPATIALEIGENVECTOREIGENVECTOR

SPATIALSPATIALEIGENVECTOREIGENVECTOR

SPATIALSPATIALEIGENVECTOREIGENVECTOR

FREEFREE

FREEFREE

FLOWFLOWCONSTR.CONSTR.

ALL FREEALL FREE

8

ISOLATING FLOW CONSTRAINTSISOLATING FLOW CONSTRAINTS

LINEAR FLOW CONSTRAINTS CAN BE EASILY DEALT WITH ,LINEAR FLOW CONSTRAINTS CAN BE EASILY DEALT WITH ,WHENEVER THEY CAN BE ISOLATED IN ONE OF THE SPATIALWHENEVER THEY CAN BE ISOLATED IN ONE OF THE SPATIALEIGENVECTORSEIGENVECTORS

AS AN EXAMPLE, THE FOLLOWING FLOW CONSTRAINT: AS AN EXAMPLE, THE FOLLOWING FLOW CONSTRAINT:

[1 1 1 … 1 1 1]

constantsector =∑i

iflowCOMPLETELY CONSTRAINS THE BEHAVIOUR OF THE SYSTEM COMPLETELY CONSTRAINS THE BEHAVIOUR OF THE SYSTEM ALONG THE FOLLOWING SPATIAL EIGENVECTOR :ALONG THE FOLLOWING SPATIAL EIGENVECTOR :

THE REMAINING SPATIAL EIGENVECTORS ARE NOT INFLUENCEDTHE REMAINING SPATIAL EIGENVECTORS ARE NOT INFLUENCEDBY THE CONSTRAINT, PROVIDED THAT THEY ORTHOGONAL TO BY THE CONSTRAINT, PROVIDED THAT THEY ORTHOGONAL TO [1[1……1].1].

9

ISOLATING FLOW CONSTRAINTSISOLATING FLOW CONSTRAINTS

FF EESECTORSECTOR

FF EESECTORSECTOR

FF EESECTORSECTOR

FF EESECTORSECTOR

FF EESECTORSECTOR

RESOURCES

RESOURCES

RESOURCES

RESOURCES

RESOURCES

RESOURCES

RESOURCES

RESOURCES

RESRES RESRES

909000 ROTATION ROTATION

FF EESECTORSECTOR

FF EESECTORSECTOR

RESRES RESRES

INVARIANT UNDER REFLECTIONINVARIANT UNDER REFLECTIONAND PERMUTATIONSAND PERMUTATIONS

BOTH EXAMPLES ADMITT A SPATIAL EIGENVECTORBOTH EXAMPLES ADMITT A SPATIAL EIGENVECTORASSOCIATED WITH THE TOTAL FLOW CONSTRAINT:ASSOCIATED WITH THE TOTAL FLOW CONSTRAINT:

constantsector =∑i

iflow

10

CONTROLLER DESIGN PROCEDURECONTROLLER DESIGN PROCEDURE

COMPUTE SPATIALCOMPUTE SPATIALEIGENVECTORSEIGENVECTORS

CHOOSE PHYSICALLYCHOOSE PHYSICALLYMEANINGFULL DECOUPLING MEANINGFULL DECOUPLING

EIGENVECTORSEIGENVECTORS

ONE OF THEONE OF THEEIGENVECTORSEIGENVECTORS

MUST BEMUST BEASSOCIATED WITHASSOCIATED WITHTHE FLOW CONST.THE FLOW CONST.

!!

DESIGN INDEPENDENTLYDESIGN INDEPENDENTLYONE CONTROLLER FORONE CONTROLLER FOR

EACH FREE SPATIAL EIGENVECTOREACH FREE SPATIAL EIGENVECTOR(NOT ASSOCIATED WITH FLOW CONSTR.)(NOT ASSOCIATED WITH FLOW CONSTR.)

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CONTROLLER IMPLEMENTATIONCONTROLLER IMPLEMENTATION

SMALL SCALESMALL SCALESYSTEMSYSTEM

SMALL SCALESMALL SCALESYSTEMSYSTEM

SMALL SCALESMALL SCALESYSTEMSYSTEM

SPATIALSPATIALEIGENVECTOREIGENVECTOR

SPATIALSPATIALEIGENVECTOREIGENVECTOR

SPATIALSPATIALEIGENVECTOREIGENVECTOR

FREEFREE

FREEFREE

FLOWFLOWCONST.CONST.

ALL FREEALL FREE

ASSUME N SMALL SCALEASSUME N SMALL SCALESYSTEMS AND ONE FLOWSYSTEMS AND ONE FLOWCONSTRAINT.CONSTRAINT.THIS LEADS TO NTHIS LEADS TO N--1 INDEPENDENT1 INDEPENDENTCONTROLLERSCONTROLLERS.

DE

CO

UPL

ING

DE

CO

UPL

ING

CONTR. 1CONTR. 1

CONTR. NCONTR. N--11 CO

UPL

ING

CO

UPL

ING

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SIMULATION EXAMPLESIMULATION EXAMPLE

FF EESECTORSECTOR

RESRES FF EESECTORSECTOR

FF EESECTORSECTOR

RESRES FF EESECTORSECTOR

RESRES

constantsector =∑i

iflow

4 SPATIAL EIGENVECTORS:4 SPATIAL EIGENVECTORS:

3 ACCOUNT FOR THE RESOURCE TRANSFER BETWEEN3 ACCOUNT FOR THE RESOURCE TRANSFER BETWEENNEIGHBORING SECTORSNEIGHBORING SECTORS

1 ACCOUNT FOR THE FLOW CONSTRAINT:1 ACCOUNT FOR THE FLOW CONSTRAINT:

13

SIMULATION EXAMPLE: SECTOR MODELSIMULATION EXAMPLE: SECTOR MODEL

⎪⎩

⎪⎨

⎧

+−−=

−+−=

⎪⎩

⎪⎨

⎧

+−−=

−+−=

•

•

•

•

dxbxbx

xaxaxax

uxbxbx

xaxaxax

ef

ef

eee

fef

eeeee

fe

fe

fff

efe

fffff

122

122111

122

122111CONTROLCONTROL(ALLOCATION (ALLOCATION DECISION)DECISION)

UNKNOWN DISTURB.UNKNOWN DISTURB.

FRIE

ND

LY

FRIE

ND

LY

EN

EM

YE

NE

MY

MAIN RESOURCE.(OUTPUT)MAIN RESOURCE.(OUTPUT)

SECONDARY RESOURCE.SECONDARY RESOURCE.

ENEMY EFFECT.ENEMY EFFECT.CONSUMPTION.CONSUMPTION.

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ENGAGING THE ENEMYENGAGING THE ENEMYENGAGING THE ENEMY

SIMULATION EXAMPLE: SETUPSIMULATION EXAMPLE: SETUP

USING 3 HUSING 3 H--INF CONTROLLERS TO MINIMIZE THEINF CONTROLLERS TO MINIMIZE THE““AMPLIFICATIONAMPLIFICATION”” FROM THE DISTURBANCE d TOFROM THE DISTURBANCE d TOA LINEAR FUNCTION OF THE MAIN RESOURCES.A LINEAR FUNCTION OF THE MAIN RESOURCES.

3 DIFFERENT OBJECTIVES WERE SIMULATED:3 DIFFERENT OBJECTIVES WERE SIMULATED:

NEUTRALIZE ENEMY PLANNEUTRALIZE ENEMY PLANNEUTRALIZE ENEMY PLAN

REDUCE INFLUENCE OF ENEMYSTRATEGIES

REDUCE INFLUENCE OF ENEMYREDUCE INFLUENCE OF ENEMYSTRATEGIESSTRATEGIES

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SIMULATION EXAMPLE: DISTURBANCESIMULATION EXAMPLE: DISTURBANCE

IN THE FOLLOWING SIMULATION SETUP, THE ENEMIEIN THE FOLLOWING SIMULATION SETUP, THE ENEMIE’’S S ALLOCATION IS REGARDED AS AN UNKNOWN DISTURBANCE.ALLOCATION IS REGARDED AS AN UNKNOWN DISTURBANCE.

THE SUBSEQUENT RESULTS WERE OBTAINED ASSUMING THETHE SUBSEQUENT RESULTS WERE OBTAINED ASSUMING THEENEMY DECIDED TO TRANSFER RESOURCES FROM SECTOR 1ENEMY DECIDED TO TRANSFER RESOURCES FROM SECTOR 1TO 4. TO 4.

0 100 200 300 400 500-1.5

-1

-0.5

0

0.5

1

1.5

0 100 200 300 400 500-1.5

-1

-0.5

0

0.5

1

1.5

0 100 200 300 400 500-1.5

-1

-0.5

0

0.5

1

1.5

0 100 200 300 400 500-1.5

-1

-0.5

0

0.5

1

1.5

SECTOR 1SECTOR 1 SECTOR 2SECTOR 2

SECTOR 3SECTOR 3

SECTOR 4SECTOR 4

disturbance.disturbance.

(Time)

RESOURCESRESOURCESLEAVINGLEAVINGSECTOR 1SECTOR 1

RESOURCESRESOURCESARRIVING ATARRIVING ATSECTOR 4SECTOR 4

ZERO ZERO REPRESENTSREPRESENTSNOMINALNOMINAL

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SIMULATION EXAMPLE: RESULTS SIMULATION EXAMPLE: RESULTS -- ENGAGING THE ENEMYENGAGING THE ENEMY

0 100 200 300 400 500-10

-5

0

5

10

0 100 200 300 400 500-10

-5

0

5

10

0 100 200 300 400 500-10

-5

0

5

10

0 100 200 300 400 500-10

-5

0

5

10

SECTOR 1SECTOR 1 SECTOR 2SECTOR 2

SECTOR 3SECTOR 3 SECTOR 4SECTOR 4

ENEMYENEMY’’S RES.S RES.

FRIEND. RES.FRIEND. RES.

(Time)

17

SIMULATION EXAMPLE: RESULTS SIMULATION EXAMPLE: RESULTS -- ENGAGING THE ENEMYENGAGING THE ENEMY

0 100 200 300 400 500-1.5

-1

-0.5

0

0.5

1

1.5

0 100 200 300 400 500-1.5

-1

-0.5

0

0.5

1

1.5

0 100 200 300 400 500-1.5

-1

-0.5

0

0.5

1

1.5

0 100 200 300 400 500-1.5

-1

-0.5

0

0.5

1

1.5

SECTOR 1SECTOR 1 SECTOR 2SECTOR 2

SECTOR 3SECTOR 3

SECTOR 4SECTOR 4

disturbance.disturbance.controlcontrol

(Time)

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SIMULATION EXAMPLE: RESULTS SIMULATION EXAMPLE: RESULTS -- ENGAGING THE ENEMYENGAGING THE ENEMY

COMMENTS:COMMENTS:

BY INSPECTION IT IS POSSIBLE TO INFER:BY INSPECTION IT IS POSSIBLE TO INFER:

•• (SLIDE 16) THAT THE FRIENDLY(SLIDE 16) THAT THE FRIENDLY’’S MAIN RESOURCE TRACKSS MAIN RESOURCE TRACKSCLOSELY THE ENEMYCLOSELY THE ENEMY’’S MAIN RESOURCE.S MAIN RESOURCE.

•• (SLIDE 17) THAT THE FRIENDLY(SLIDE 17) THAT THE FRIENDLY’’S ALLOCATION IS VERYS ALLOCATION IS VERYSIMILAR TO THE ENEMYSIMILAR TO THE ENEMY’’S ALLOCATION. THIS SHOWS THAT FROMS ALLOCATION. THIS SHOWS THAT FROMTHE MEASUREMENTS OF MAIN RESOURCES, THE CONTROLLER IS THE MEASUREMENTS OF MAIN RESOURCES, THE CONTROLLER IS ABLE TO FIGURE OUT WHAT WAS THE ENEMYABLE TO FIGURE OUT WHAT WAS THE ENEMY’’S STRATEGY AND ACTS STRATEGY AND ACTACCORDINGLY.ACCORDINGLY.

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SIMULATION EXAMPLE: RESULTS SIMULATION EXAMPLE: RESULTS -- ENGAGING THE ENEMYENGAGING THE ENEMY

0 100 200 300 400 500-1.5

-1

-0.5

0

0.5

1

1.5

CONTROLLER 1CONTROLLER 1

0 100 200 300 400 500-1.5

-1

-0.5

0

0.5

1

1.5

CONTROLLER 2CONTROLLER 2

0 100 200 300 400 500-1.5

-1

-0.5

0

0.5

1

1.5

CONTROLLER 3CONTROLLER 3

(Time)

DECOUPLED CONTROLLERSDECOUPLED CONTROLLERSFLUX FROM SECTOR 1 TO 2FLUX FROM SECTOR 1 TO 2

FLUX FROM SECTOR 2 TO 3FLUX FROM SECTOR 2 TO 3 FLUX FROM SECTOR 3 TO4FLUX FROM SECTOR 3 TO4

20

0 100 200 300 400 500-10

-5

0

5

10

0 100 200 300 400 500-10

-5

0

5

10

0 100 200 300 400 500-10

-5

0

5

10

0 100 200 300 400 500-10

-5

0

5

10

SECTOR 1SECTOR 1 SECTOR 2SECTOR 2

SECTOR 3SECTOR 3

SECTOR 4SECTOR 4

ENEMYENEMY’’S RES.S RES.

FRIEND. RES.FRIEND. RES.

(Time)

SIMULATION EXAMPLE: RESULTS SIMULATION EXAMPLE: RESULTS -- ENEMY REDUCTIONENEMY REDUCTION

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SIMULATION EXAMPLE: RESULTS SIMULATION EXAMPLE: RESULTS -- FRIEND. REDUCTIONFRIEND. REDUCTION

0 100 200 300 400 500-20

-10

0

10

20

SECTOR 1SECTOR 1

0 100 200 300 400 500-20

-10

0

10

20

SECTOR 2SECTOR 2

0 100 200 300 400 500-20

-10

0

10

20

SECTOR 3SECTOR 3

ENEMYENEMY’’S RES.S RES.

FRIEND. RES.FRIEND. RES.

0 100 200 300 400 500-20

-10

0

10

20

SECTOR 4SECTOR 4

(Time)