evolving systems: an introduction · s. a. frost and m. j. balas, “evolving systems: adaptive key...
TRANSCRIPT
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Evolving Systems:An Introduction
Mark BalasECE Department
University of WyomingLaramie, WY
with help fromSusan Frost
Intelligent Systems DivisionNASA Ames Research Center
Devolving System
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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
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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
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Evolved System
How It Works
Controller 1 Controller 2
Mating
Controller 1 Controller 2
Susan’s Slide
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Evolved System
ActiveComponent 1
ActiveComponent 2
Mated Components
ActiveComponent 1
ActiveComponent 2
The Process of Evolving Systems
ActiveComponent 3
ActiveComponent 3
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It’s not theories about stars; it’s the actual stars that count.”……… Freeman Dyson
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International SpaceStation after
9 December 2006Mission
Evolving Systems Applications
Autonomous Assembly in Space
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Evolving Systems Applications
Autonomous Rendezvous and Docking
Servicing and System Upgrades
DARPA’s OrbitalExpress
(ASTRO ServicingSatellite pictured on left)
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Stability is Essential During theEntire Evolution Process
Orion Crew Exploration Vehicle Docking with the ISS
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Launch Vehicles: DevolvingSystems
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NASA-MSFC Ares-Orion
$$$
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Constellations and Formations ofSpacecraft (NASA-JPL)
Separated Spacecraft Interferometers
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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
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EvolvingSpacecraft Formations
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Joining Spacecraft
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Smart Grids:Virtual Interconnecting Forces
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“It is surprising how quickly we replace a human operatorwith an algorithm and call it SMART”
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Affine Nonlinear Systems
15
)(
)()(
xCy
uxBxAx
With smooth vector fields:defined on a neighborhood a smooth manifold, or
))(),(),(( xCxBxAN
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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
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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
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Evolving Systems:Evolution=Homotopy
Evolved System
10
ActiveComponent 1
Controller 1
ActiveComponent 2
Controller 2
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Genetics of EvolvingSystems:Inheritance ofComponent Traits
Controllability/Observability StabilityDissipativity Optimality Robustness Disturbance Rejection/Signal Tracking
Source: CNN.com
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How To Analyze Evolving Systems
Admittance/Impedance
Perturbation Methods
Graph Theory Differential Geometry- Lie GroupsOther stuff
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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
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Fully Evolved System
Component 1
Component 212 xx
21 ff 11 xv
)( 111 fYv
)( 222 vZf
2f
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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
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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.
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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
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Dissipativity:“Higher” Form of Stability
System
u y
SpousesRoommates/
CapacitorsandInductors
MassesandSpringse.g.
SuppliedEnergyStoredEnergy
System
u y
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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
])()([)(/))((
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Strict Dissipativity
)()(
)()(
wheneDissipativStrictlyisSystem
0)0(
;0;0)(:FunctionStorageEnergy
)(
)(
xCxVBVL
xSxVAVL
V
xxV
TxB
xA
InternalPower
Dissipated
)(,)()(
])()([)(/))((
ratestorageenergy
xSuyuxCxS
uxBxAVtxVdttxdV
T
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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
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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
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Almost Strictly Dissipative (ASD)
Gy
Output FB
+
NonlinearSystemu Strictly
Dissipativey
)(
)()(
xCy
uxBxAx
ASDis))(),(),()()()(( xCxBxGCxBxAxAC
For LTI Systems: ASD=ASPR
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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)()(
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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
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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!
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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:
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System2
Partial or Failed Evolutions:Evolving Systems: Completely Connected
System1
Subsystem 3
System1
Subsystem 3
Evolved System
System2
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System1
System2
Evolving Systems: Failure to Connect
Subsystem 3
Subsystem 3Should connectto System 2, BUTdid not!!!!!!
System1 System2
Subsystem 3
Evolved System with Failure
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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
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System with Key ComponentEvolving Controller
Closed Loop Poles of Example
No Evolving Controllers
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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
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ADAPTIVE Key ComponentEvolving Controller
Controller 1
…ADAPTIVEKey ComponentController
Component NComponent 2
Component 1Key
Component
Controller 2 Controller N
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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
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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
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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
![Page 45: Evolving Systems: An Introduction · S. A. Frost and M. J. Balas, “Evolving systems: Adaptive key component control and inheritance of passivity and dissipativity,” Applied mathematics](https://reader035.vdocuments.site/reader035/viewer/2022062508/6047e9988849c114403a14cc/html5/thumbnails/45.jpg)
45
A Theorem Is WorthA Thousand Simulations
(despite its limitations)
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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
![Page 47: Evolving Systems: An Introduction · S. A. Frost and M. J. Balas, “Evolving systems: Adaptive key component control and inheritance of passivity and dissipativity,” Applied mathematics](https://reader035.vdocuments.site/reader035/viewer/2022062508/6047e9988849c114403a14cc/html5/thumbnails/47.jpg)
Now Let’s See Some DetailedMathematical Proofs
No No Please, I’d Rather Be EatenAlive by Radioactive Spiders
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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
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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
![Page 50: Evolving Systems: An Introduction · S. A. Frost and M. J. Balas, “Evolving systems: Adaptive key component control and inheritance of passivity and dissipativity,” Applied mathematics](https://reader035.vdocuments.site/reader035/viewer/2022062508/6047e9988849c114403a14cc/html5/thumbnails/50.jpg)
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
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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
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EvolvingSpacecraft Formations
52
Are You Lookin’ at Me?
Joining Spacecraft
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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
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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
![Page 55: Evolving Systems: An Introduction · S. A. Frost and M. J. Balas, “Evolving systems: Adaptive key component control and inheritance of passivity and dissipativity,” Applied mathematics](https://reader035.vdocuments.site/reader035/viewer/2022062508/6047e9988849c114403a14cc/html5/thumbnails/55.jpg)
Spacecraft Dynamics
55
DkDkD
kDk
Dk
Dk
Dkkkkk
LzFzz
zu
NkuBuAxxSC
or:GeneratoreDisturbanc
,...,1;:
IdenticalSpacecraft
1.01,1
0,
00
10IntegratorDouble
CBA
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Open Chain Formation
56
SC1
SC2
SC3SC4
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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
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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
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Open Chain Formation withReciprocity
59
SC1
SC2
SC3SC4
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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?
![Page 61: Evolving Systems: An Introduction · S. A. Frost and M. J. Balas, “Evolving systems: Adaptive key component control and inheritance of passivity and dissipativity,” Applied mathematics](https://reader035.vdocuments.site/reader035/viewer/2022062508/6047e9988849c114403a14cc/html5/thumbnails/61.jpg)
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
![Page 62: Evolving Systems: An Introduction · S. A. Frost and M. J. Balas, “Evolving systems: Adaptive key component control and inheritance of passivity and dissipativity,” Applied mathematics](https://reader035.vdocuments.site/reader035/viewer/2022062508/6047e9988849c114403a14cc/html5/thumbnails/62.jpg)
Open Chain Formation
62
SC1
SC2
SC3SC4
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Theorem:
63
stable2and
stable
20
0
00
BGCABGCA
BGCABGC
BGCABGC
BGCA
A
CA
C
0stable1.01,1
0,
00
10.
gGACBACor C
![Page 64: Evolving Systems: An Introduction · S. A. Frost and M. J. Balas, “Evolving systems: Adaptive key component control and inheritance of passivity and dissipativity,” Applied mathematics](https://reader035.vdocuments.site/reader035/viewer/2022062508/6047e9988849c114403a14cc/html5/thumbnails/64.jpg)
Closed Chain Formation
64
SC1
SC2
SC3SC4
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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
![Page 66: Evolving Systems: An Introduction · S. A. Frost and M. J. Balas, “Evolving systems: Adaptive key component control and inheritance of passivity and dissipativity,” Applied mathematics](https://reader035.vdocuments.site/reader035/viewer/2022062508/6047e9988849c114403a14cc/html5/thumbnails/66.jpg)
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
![Page 67: Evolving Systems: An Introduction · S. A. Frost and M. J. Balas, “Evolving systems: Adaptive key component control and inheritance of passivity and dissipativity,” Applied mathematics](https://reader035.vdocuments.site/reader035/viewer/2022062508/6047e9988849c114403a14cc/html5/thumbnails/67.jpg)
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
![Page 68: Evolving Systems: An Introduction · S. A. Frost and M. J. Balas, “Evolving systems: Adaptive key component control and inheritance of passivity and dissipativity,” Applied mathematics](https://reader035.vdocuments.site/reader035/viewer/2022062508/6047e9988849c114403a14cc/html5/thumbnails/68.jpg)
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
![Page 69: Evolving Systems: An Introduction · S. A. Frost and M. J. Balas, “Evolving systems: Adaptive key component control and inheritance of passivity and dissipativity,” Applied mathematics](https://reader035.vdocuments.site/reader035/viewer/2022062508/6047e9988849c114403a14cc/html5/thumbnails/69.jpg)
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
![Page 70: Evolving Systems: An Introduction · S. A. Frost and M. J. Balas, “Evolving systems: Adaptive key component control and inheritance of passivity and dissipativity,” Applied mathematics](https://reader035.vdocuments.site/reader035/viewer/2022062508/6047e9988849c114403a14cc/html5/thumbnails/70.jpg)
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*
![Page 71: Evolving Systems: An Introduction · S. A. Frost and M. J. Balas, “Evolving systems: Adaptive key component control and inheritance of passivity and dissipativity,” Applied mathematics](https://reader035.vdocuments.site/reader035/viewer/2022062508/6047e9988849c114403a14cc/html5/thumbnails/71.jpg)
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
![Page 72: Evolving Systems: An Introduction · S. A. Frost and M. J. Balas, “Evolving systems: Adaptive key component control and inheritance of passivity and dissipativity,” Applied mathematics](https://reader035.vdocuments.site/reader035/viewer/2022062508/6047e9988849c114403a14cc/html5/thumbnails/72.jpg)
Wonderful New Survivable System
A Sortof Paradigm
72
Old DecrepitBroken-downSystem(eg USpower grids)
NewExcitingAdaptiveController