1

Evolving Systems:An Introduction

Mark BalasECE Department

University of WyomingLaramie, WY

with help fromSusan Frost

Intelligent Systems DivisionNASA Ames Research Center

Devolving System

2

References Balas, M. J., Frost, S. A., and Hadaegh, F. Y., “Evolving Systems: A Theoretical Foundation,”

Proceedings of AIAA Guidance, Navigation, and Control Conference, Keystone, CO, 2006.

Balas, M. J., and Frost, S. A., “An Introduction to Evolving Systems of Flexible AerospaceStructures,” Proceedings IEEE Aerospace Conference, Big Sky, MT, 2007.

Balas, M. J. and Frost, S. A., “Evolving Systems: Inheriting Stability with Evolving Controllers”,Proceedings 47th Israel Annual Conference on Aerospace Sciences, Tel-Aviv, Israel, 2007.

Frost, S. A. and Balas, M. J., “Stability Inheritance and Contact Dynamics of Flexible StructureEvolving Systems”, Proceedings 17th IFAC Symposium on Automatic Control in Aerospace,Toulouse, France, 2007.

Frost, S. A. and Balas, M. J., “Stabilizing Controllers for Evolving Systems with Application toFlexible Space Structures”, AIAA Guidance, Navigation, and Control Conference, Hilton Head,South Carolina, 2007.

Frost, S. A and Balas, M. J., “Adaptive Key Component Control and Inheritance of Almost StrictPassivity in Evolving Systems", AAS Landis Markley Symposium, Cambridge, MD, 2008

Frost, S. A. and Balas, M. J., “Adaptive Key Component Controllers for Evolving Systems”, AIAAGuidance, Navigation, and Control Conference, Honolulu, HI, 2008 .

Balas, M. J., and Frost, S.A., “Adaptive Key Component Control of Nonlinear Evolving FlexibleStructures", ASME Smart Materials, Adaptive Structures and Intelligent Systems, Ellicott City, MD,2008

S. A. Frost and M. J. Balas, “Evolving systems: Adaptive key component control and inheritance ofpassivity and dissipativity,” Applied mathematics and Computation, May 2010

3

What It Is?Evolving Systems=AutonomouslyAssembledActive Structures

Or Self-AssemblingStructures,which Aspire to aHigher Purpose;Cannot be attainedby Components Alone

Control

EstimationModeling& Dynamics

Guidance

Control

EstimationModeling& Dynamics

Guidance

Control

EstimationModeling& Dynamics

Guidance

Control

EstimationModeling& Dynamics

Guidance

Control

EstimationModeling& Dynamics

Guidance

Control

EstimationModeling& Dynamics

Guidance

4

Evolved System

How It Works

Controller 1 Controller 2

Mating

Controller 1 Controller 2

Susan’s Slide

5

Evolved System

ActiveComponent 1

ActiveComponent 2

Mated Components

ActiveComponent 1

ActiveComponent 2

The Process of Evolving Systems

ActiveComponent 3

ActiveComponent 3

It’s not theories about stars; it’s the actual stars that count.”……… Freeman Dyson

7

International SpaceStation after

9 December 2006Mission

Evolving Systems Applications

Autonomous Assembly in Space

8

Evolving Systems Applications

Autonomous Rendezvous and Docking

Servicing and System Upgrades

DARPA’s OrbitalExpress

(ASTRO ServicingSatellite pictured on left)

9

Stability is Essential During theEntire Evolution Process

Orion Crew Exploration Vehicle Docking with the ISS

Launch Vehicles: DevolvingSystems

10

NASA-MSFC Ares-Orion

$$$

11

Constellations and Formations ofSpacecraft (NASA-JPL)

Separated Spacecraft Interferometers

12

Global Persistent Surveillance:Constellations-Formations-Evolving Systems (NRO/DARPA)The next step in network-centric warfare will be the creation ofnetworked sensing suites that tailor their observations to theadversary’s rate of activity. These various sensors will concentrateon observing changes rather than on observing scenery …

Signal Magazine

Evolution

Devolution

ConstellationsFormations Mated Systems

EvolvingSpacecraft Formations

13

Joining Spacecraft

Smart Grids:Virtual Interconnecting Forces

14

“It is surprising how quickly we replace a human operatorwith an algorithm and call it SMART”

Affine Nonlinear Systems

15

)(

)()(

xCy

uxBxAx

With smooth vector fields:defined on a neighborhood a smooth manifold, or

))(),(),(( xCxBxAN

16

Evolving Systems: General

)(

)()(

Componentthi

iii

ciiiiii

xCy

uxBxAx

nx

x

x

xxCy

uxBxAx 2

1

withSystemEvolved)(

)()(

jiij

ij

ctionsInterconne

jjiij

L

jijiiiiii uxxAuxBxAx

10;),,()()(where

1

),,(:ControllerLocal

iiiii

iiici

uyzlz

u)(zhu

17

Evolving Systems: 2 Components

)(

),,()()(

2Component

222

11221222222

xCy

uxxAuxBxAx

)(

),,()()(

1Component

111

22112111111

xCy

uxxAuxBxAx

)()(

)(

),,(

),,(

)(0

0)(

)(

)(

SystemEvolved

22

11

11221

22112

)(

22

11

)(

22

11

xCxC

xCy

uxxA

uxxAu

xB

xB

xA

xAx

xBxA

connectedfully1

dunconnecte0

10 2112

18

Evolving Systems:Evolution=Homotopy

Evolved System

10

ActiveComponent 1

Controller 1

ActiveComponent 2

Controller 2

19

Genetics of EvolvingSystems:Inheritance ofComponent Traits

Controllability/Observability StabilityDissipativity Optimality Robustness Disturbance Rejection/Signal Tracking

Source: CNN.com

20

How To Analyze Evolving Systems

Admittance/Impedance

Perturbation Methods

Graph Theory Differential Geometry- Lie GroupsOther stuff

21

Component 1 Component 2

21

211

xxxxv

12 ff

11,

)(:OperatorAdmittance

)(:OperatorImpedance

ZYYZ

FYVY

VZFZ

Linear Case: C. M. Harris and A. G. Piersol,Shock and Vibration Handbook, McGraw-Hill,NY , 2001

Impedance and Admittance of Components

22

Fully Evolved System

Component 1

Component 212 xx

21 ff 11 xv

)( 111 fYv

)( 222 vZf

2f

23

Nonlinear State Space Version: Admittance-Impedance

12 yu

21 uu

)(

)(

)()()(

111

111

111111111

xCy

xCy

uxBuxBxAx

AA

AA

)(

)(

)()()(

222

222

222222222

xCy

xCy

uxBuxBxAx

AA

AA

Component 1

Component 2

Au1

Au2

Ay1

Ay2

)( 111 uYy

)( 222 uZy

System 1

System 2

111 fYv

222 vZf

Characterize LINEAR Evolved System Stability

Q: Given System 1, can we find all System 2 so thatconnected system is stable ?Use Youla Parametrization for nonproper Impedances/Admittances.

Component 1

Youla or Q Parameterization

)()( 12 QDXQNYZ

stableandcoprime

arewhere

11

N,D

uNDuY

function.rationalstableanybeand,and

stable,andcoprimebeLetQIDYNX

X,Y

Component 2All controllersthat make theclosed-loopstable.

M. Vidyasagar,Control System Synthesis:A Factorization Approach,MIT Press,1985

26

Dissipativity:“Higher” Form of Stability

System

u y

SpousesRoommates/

CapacitorsandInductors

MassesandSpringse.g.

SuppliedEnergyStoredEnergy

System

u y

27

Definition of Dissipativity

)()(

0)(

wheneDissipativisSystem

0)0(

;0;0)(:FunctionStorageEnergy

)(

)(

xCxVBVL

xVAVL

V

xxV

TxB

xA

System

)(

)()(

xCy

uxBxAxu y

ExternalPower

ExternalEnergy

duyxVtxV

uytutyuxC

uxBxAVtxVdttxdV

t

edEnergyStorInitialedEnergyStor

TT

0

ratestorageenergy

,))0(())((

or

,)()()(0

])()([)(/))((

28

Strict Dissipativity

)()(

)()(

wheneDissipativStrictlyisSystem

0)0(

;0;0)(:FunctionStorageEnergy

)(

)(

xCxVBVL

xSxVAVL

V

xxV

TxB

xA

InternalPower

Dissipated

)(,)()(

])()([)(/))((

ratestorageenergy

xSuyuxCxS

uxBxAVtxVdttxdV

T

Positive Realness (PR)

T

TT

CPB

QLLPAPA

DBAsICsT

CBA

0Y-K

thatso0PisThere

:)()(and

),,(Given1

)(RealPositiveof

definitiontheastakenoftenisThis

allfor02/))()(()(Re *

PR

jTjTjT

A FrequencyDomain

Condition

Linear&Dissipative

For Linear Systems: PR= Dissipative= Passive

What does Strictly Positive Real (SPR) mean?

Answer: Lotsa Things, Not all equivalent!! (See J.Wen,AC-33,1988)Here is ONE Definition of SPR:

PRis2/))()-(()-(Re

thatso0isThere*

jTjTjT

T

TT

CPB

QPLLPAPA

IAA

02

Y-KintoChangeJust

:Y-KoRelation t

Linear &StrictlyDissipative

For Linear Systems: SPR=Strictly Dissipative=Strictly Passive

31

Almost Strictly Dissipative (ASD)

Gy

Output FB

+

NonlinearSystemu Strictly

Dissipativey

)(

)()(

xCy

uxBxAx

ASDis))(),(),()()()(( xCxBxGCxBxAxAC

For LTI Systems: ASD=ASPR

32

Linear ASPR via Non-OrthogonalProjections

2) Almost PR if and only ifCB positive definite,open-loop transfer function is weakly minimum phase(i.e. can have some marginally stable transmission zeros),andmarginally stable zero dynamics are PR

Balas&Fuentes :1) (A,B,C) Almost SPR if and only if CB positive definite and

open-loop transfer functionis minimum phase

( i.e. all transmission zeros stable)

BAsICsP 1)()(

33

ASD

Inheritance of Almost Strictly Dissipative Property

12 yu

21 uu

)(

)(

)()()(

111

111

111111111

xCy

xCy

uxBuxBxAx

AA

AA

)(

)(

)()()(

222

222

222222222

xCy

xCy

uxBuxBxAx

AA

AA

Component 1

Component 2

Au1

Au2

Ay1

Ay2

ASD

ASD

34

1k1f

m1

1u

1

11

xxv

Component 1

Two Component Flexible StructureEvolving System

Contactpoint

Component 2

3

33

xxv

m2

2

22

xxv

m3

3u

2k

2u

2f

Component 2

3

33

xxv

m2

2

22

xxv

m3

3u

2k

2u

2f

Example wherestability is lost duringevolution will follow!

35

System That Does Not InheritStability

1and4,1,1,30,10with 2312321 kkmmm

.

ComponentControllers:

T

333

T

222

2323333

12123223222

,

,

)(

)()(

qqy

qqy

qqkuqm

qqkqqkuqm

T

111

2112111

,

)(

qqy

qqkuqm

33

22

11

16.0

5.02.01.0

1.09.0

qsu

qss

u

qsu

Component 1:

Component 2:

36

System2

Partial or Failed Evolutions:Evolving Systems: Completely Connected

System1

Subsystem 3

System1

Subsystem 3

Evolved System

System2

37

System1

System2

Evolving Systems: Failure to Connect

Subsystem 3

Subsystem 3Should connectto System 2, BUTdid not!!!!!!

System1 System2

Subsystem 3

Evolved System with Failure

38

Controller 1

…EvolvingController

Component NComponent 2

Component 1Key

Component

Controller 2 Controller N

Key Component Evolving Controller

Local Controllers 1 … N and theircomponents’ input-output ports remainunchanged

Functionof epsilon

39

System with Key ComponentEvolving Controller

Closed Loop Poles of Example

No Evolving Controllers

40

Benefits of Key ComponentEvolving Controllers

Maintains stability during Evolution

Interchangeability of non-key components

Cost savings & risk mitigation

No clear design methodology

Controller 1

…EvolvingController

Component NComponent 2

Component 1Key

Component

Controller 2 Controller N

41

ADAPTIVE Key ComponentEvolving Controller

Controller 1

…ADAPTIVEKey ComponentController

Component NComponent 2

Component 1Key

Component

Controller 2 Controller N

42

Direct Adaptive Model Following Control(Wen-Balas 1989)

0tye

Plant

Reference

Model

Gm

Gu

Ge

um ym

xm

yu

+

Adaptive

Gain Laws

SignalsKnown),,( ymm exu

43

Direct Adaptive Persistent Disturbance Rejection(Fuentes-Balas 2000)

0tye

Plant

Reference

model

Ge

um ym

xm

yu

+

Adaptive

Gain Laws

SignalsKnown),,,(BasiseDisturbanc

Dymm exu

Disturbance

Generator

44

Adaptive Key Component Control

AdaptiveGain

ASD

12 yu

21 uu

)(

)(

)()()(

111

111

111111111

xCy

xCy

uxBuxBxAx

AA

AA

)(

)()(

222

222222

xCy

uxBxAx

Key Component 1

All Other Components

Au1

Ay1

Strictly Dissipative

45

A Theorem Is WorthA Thousand Simulations

(despite its limitations)

46

Adaptive Key Component Controller Theorem

),( 11AA yuIf Key Component 1 is Almost Strictly Dissipative and

Component 2 is Strictly Dissipative, then)( 11

AA yu

0;)( 11111

111

TAA

AA

yyG

yGu

AdaptiveKey ComponentController

),( 22 yu

Produces boundedisand0, 121 Gxxt

throughout the entire evolution 10

Now Let’s See Some DetailedMathematical Proofs

No No Please, I’d Rather Be EatenAlive by Radioactive Spiders

48

edissipativis2System,)(

))((

02

1Let

2211

111111

1112

2

uyyGy

GyytraceGGtraceV

GGtraceV

ATA

TTAAT

T

Adaptive Gain Law

Strictly DissipativeSystem

11111 )( TAA yyGG

Ayu 12

21 yu A Ay1

AyGy 112

1G

System 1

System 2

49

Adaptive Key Component Control Can Mitigate PersistentDisturbances Also

AdaptiveGain

12 yu

21 uu

)(

)(

)()()(

111

111

11111111111

xCy

xCy

uuxBuxBxAx

AA

eDisturbancPersistent

DAA

)(

)()(

222

222222

xCy

uxBxAx

Key Component 1

All Other Components

Au1

Ay1

50

Adaptive Key Component Controlof Nonlinear Plant

0 20 40 60 80 100 120 140 160 180 200-150

-100

-50

0

50

100

150

Time (seconds)

Positio

n

Mass Displacement with No Adaptive Control

Mass 1

Mass 2

Mass 3

0 20 40 60 80 100 120 140 160 180 200-8

-6

-4

-2

0

2

4

6

8

10

Time (seconds)

Positio

n

Mass Displacement with Adaptive Control on Mass 1

Mass 1

Mass 2

Mass 3

51

Adaptive Key ComponentController

Component 1 is Key Component w/ I/Oports on mass 1

Component 2 is Key Component w/ I/Oports on mass 2

Nonminimum phase zeros:0.0051466 + 0.20089i,0.0051466 - 0.20089i

EvolvingSpacecraft Formations

52

Are You Lookin’ at Me?

Joining Spacecraft

Control Based on Relative State

53

01121 t

rxx

12 Gu

damperpreloaded

112

springpreloaded

1P12P

112

112

12

)()rq(qg DD

D

PDP

rqqg

rqq

rqqgg

Gu

SC2 SC1

Uni-directionalNewton’s 3rdLaw

Reciprocity= Usual Newton’s Laws

54

01121 t

rxx

12 Gu

SC2 SC1

damperpreloaded

112

springpreloaded

1P12P

112

)()rq(qg DD rqqg

uGu

211 uGu

Spacecraft Dynamics

55

DkDkD

kDk

Dk

Dk

Dkkkkk

LzFzz

zu

NkuBuAxxSC

or:GeneratoreDisturbanc

,...,1;:

IdenticalSpacecraft

1.01,1

0,

00

10IntegratorDouble

CBA

Open Chain Formation

56

SC1

SC2

SC3SC4

Relative Measurements

57

)(: 111 kkkkkk rxxCCySC

DkDkkkkkkk

DkDk

DkDkk

zGGCyyGuuu

u

zGGCzGGyu

)()(

0:FeedbackOutputChained

111

1

1

11

)()(or0

1

)(&

2;)()(

BGCA

kzBGBGCBGCA

uuBADk

BRR

Dkk

Dkkkk

Open Chain Stability

58

stablestable

0

0

00

Let

)(

3

2

1

1

BGCAA

BGCABGC

BGCABGC

BGCA

BGCBGCA

C

A

kkk

C

4 S/C

Open Chain Formation withReciprocity

59

SC1

SC2

SC3SC4

Stability Open Chain withReciprocity

60

DD

DD

DD

zGGCu

GCzGGCu

zGGCu

u

434

3323

212

1

)(

0

:FeedbackOutput

Reciprocity

??andstable

20

00

Let

3

2

1

BGCAA

BGCABGC

BGCBGCABGC

BGCA

C

AC

Can it be Unstable?

Theorem: Open Chain with Reciprocity

61

)01(or0.62,-1.6252

1

2

1where

stable)1(,)2(and

stable

20

00

2

BGCABGCABGCA

BGCABGC

BGCBGCABGC

BGCA

A

CA

C

#)1(

)1()2(

0

2

0:Proof

2

BGCABGC

BGCBGCA

I

II

BGCABGC

BGCBGCA

I

II

0stable1.01,1

0,

00

10.

gGACBACor C

Open Chain Formation

62

SC1

SC2

SC3SC4

Theorem:

63

stable2and

stable

20

0

00

BGCABGCA

BGCABGC

BGCABGC

BGCA

A

CA

C

0stable1.01,1

0,

00

10.

gGACBACor C

Closed Chain Formation

64

SC1

SC2

SC3SC4

65

stable.is)(,0,0

000

000

0

0

00

0

0

)1(

)(

00

stable2andstable0

C

ABGCABGCAA

C

A

BGCBGCBGC

BGCABGC

BGCABGC

BGCA

BGCABGC

BGCABGC

BGCBGCBGCA

A

1forunstableand),10~theory(02.00stable)(

1.01,1

0,

00

10.

13-0

CA

CBACor

66

ADAPTIVE Key ComponentController

Controller 1

SC 1Key

Component Component 2

Controller 2

Component N

Controller N

SC 2

SC NADAPTIVEKey ComponentController

Was Unstable, butAdaptive Key Component Controller Restores Stability

Example

67

)1

10()(

1.01,1

0,

00

10:IntegratorDouble

ssC

CBA

0 5 10 15 20 25 30 35 40 45 50-60

-40

-20

0

20

40

60

Time (sec)

E3

Response

Key Component Adaptive Control

E3=x4-x3-r3

0 5 10 15 20 25 30 35 40 45 50-30

-20

-10

0

10

20

30

40

Time (sec)

E4

Re

sp

on

se

Key Component Adaptive Control

E4=x4-x1-r1

44

4

24

4

4

34

~:OutputMeasured

1;~

100;)~(GainsAdaptive

1;~ControllerAdaptiveComponentKey

Cxy

yG

yG

GyGGCu

eeDD

eee

D

ControlAdaptive

DDe

ControlSCOriginal

0

16;

0

8;

0

5;

0

34321 rrrr

Conjecto-Theorem

68

thenphase,minimumis)(P(s)and0CBie

ASPR,areC)B,(A,dynamicsspacecraftindividualIf1 BAsIC

NN

eeDND

eeNe

D

ControlAdaptive

DDNe

ControlNSCOriginal

NN

Cxy

yG

yG

GyGGCu

~:OutputMeasured

1;~

100;)~(GainsAdaptive

bounded;~)spacecraftjoining(on theControllerAdaptiveComponentKey

2

1

gainsadaptiveboundedwith

es)disturbancboundedrejects(and

0produces 1 krxxtkkkk

Proof:

69

CGBA

C

C

C

C

C

GG

GG

GGG

B

B

B

A

A

A

BGCABGC

BGCABGC

BGCBGCBGCA

A

CxCy

uBA

00

00

00

0

0

)1(

00

00

00

00

00

00

0

0

)1(

)(where

~~

~)(

Let

)(

444

4

3

2

1

WLOG use 4 SC

70

#)~

,~

),((),,(

phase.minimum)(phaseminimum)(and

00Clearly

ASPRCBAASPRCBAR(A,B,C)ASP

sPsP

CBBC

C

CGGBA

C

C

C

IG

IB

B

B

C

C

C

GG

GG

GGG

B

B

B

A

A

A

CG

B

AA

yGuG

G

CGB

CGBA

CC

)(

00

00

00

000

0

00

00

00

)

00

00

00

0

0

)1(

00

00

00

00

00

00

(

000

0

)()(~

~Find

*

*

~

*

*

)(

*

4*4*

Future Formation Stuff

71

•Nonlinear:

•Harder but Doable

•Maintaining a Formation Shape (Tracking); Immutability ofFormation Shapes

•Swarms: George Hill’s Eqs(or Johnny-Come-Lately: Clohessy-Wiltshire)

kk

kkkk

Cxy

BuxgAxx

enoughweak&Lipschitz

)(

)(

)()(

kk

kkkk

xCy

uxBxAx

Wonderful New Survivable System

A Sortof Paradigm

72

Old DecrepitBroken-downSystem(eg USpower grids)

NewExcitingAdaptiveController