cours sci31_reine talj_a2015 - séances 1 et 2
TRANSCRIPT
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Modeling, control and observation of
dynamical systems – case of Systems
of Systems (SoS)
Ali Charara
Reine Kfoury [email protected]
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!rogram• Unified representation, examples of modeling
• Analysis of systems properties : stability, controllability, observability, …
• Linearization
• Linear systems : control and observer – State feedback – Luenberger observers – Input-output decoupling control
• Nonlinear control
– Global linearization (input-output decoupling) – Lyapunov
• Introduction to Systems of Systems control – Hierarchical control
– Decentralized and Distributed control – Networked control
Examples and case study (Matlab)
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Multimodal trans"ort services
Syst#me de Syst#mes$%am"les of Systems of Systems
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$&'ealth
Syst#me de Syst#mes$%am"les of Systems of Systems
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atural disasters management
Syst#me de Syst#mes$%am"les of Systems of Systems
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$%am"les of Systems of Systems
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$%am"les of Systems of Systems
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$%isting Com"le% Systems$%clusive, Autonomous, ocal
Transformation
(Keating, et al., 2003)
*hat are Systems of Systems+
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Integrated, Aligned, and
Transforming
System of Systems
nterconnected, ntegrated Mission,
-lobal, $mergent Structure
(Keating, et al., 2003)
*hat are Systems of Systems+
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A schematic re"resentation of a system of systems
(Samad, Parisini, Part3-IEEE CSS)
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Some Definitions for « Systems of Systems (SoS) »
• Systems of Systems are large-scale integrated systems which areheterogeneous and independently operable on their own, but
are networked together for a common goal . The goal may be
cost, performance, robustness, etc…
• A System of Systems is a "super system" comprised of other elements which themselves are independent complex
operational systems and interact among themselves to achievea common goal . ach element of a SoS achieves well-
substantiated goals even if they are detached from the rest of
the SoS.
(Karcanias, 2011)
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Characteristics of Systems of Systems
• Operational independence of component systems
• Managerial independence of component systems
• eographical distri!ution
• "mergent !eha#ior
• "#olutionary de#elopment processes
(Maier$ %&&')
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Some a""lication domains of SoS
• ir raffic *ontrol
• +nternet
• +ntelligent ransport Systems
• ,ene-a!le energy systems
• ,o!otic s-arms
• Space
• Defense and military• "n#ironment$ etc
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arge Scale Systems vs Systems of Systems
/SS
…
Traditional LSS Modeling
/SS
TOP
BOT.
BOT.
TOP
SoSE Modeling Difficulty
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An introduction to Systems of Systems Control
!ierarchical #ontrol
$ecentrali%ed #ontrol
$istributed #ontrol
#onsensus-based #ontrol
&etworked #ontrol
Modeling. 0ard to find a simple mathematical formalism to define all systems
aspects.
Control. "#ery case ha#e its o-n study and la-.
Communication. Systems operate using different languages and semantics.
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An introduction to Systems of Systems Control
0ierarchical control
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An introduction to Systems of Systems Control
0ierarchical control
A system is chosen to play the role of a coordinator
ach system receives its &-' neighbors data via wireless communication
An optimal controller is designed for each system
The coordination of the & solutions via an iterative algorithm gives an
optimal solution of the SoS
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*entrali1ed control Decentrali1ed control
An introduction to Systems of Systems Control
Ta/en from the theory of large&scale (com"le%) systems, one can
share the control action among a finite number of local controllers0
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An introduction to Systems of Systems Control
Decentrali1ed control
The sensory information of SoS is distributed across the & domains & local controllers are designed to meet systems( criteria
A global control component reacting to neighboring )&-'* systems is
added to the local control strategies.
A cost function can be chosen for the local design problems
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Distri!uted control
An introduction to Systems of Systems Control
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An introduction to Systems of Systems Control
*onsensus2!ased *ontrol
+t is a cooperati#e control paradigm !ased on 3consensus4 among
systems in a SoS.
Multiple ro#ers
5o single control unit6ro#er
7nits must agree on goal Su! goals may !e different for each unit
Shared communications
(,en and 8aird 9::')
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An introduction to Systems of Systems Control
*onsensus2!ased *ontrol
*ommunications
0o- -ell can the ro!ots tal; to each other % cannot tal; to ? directly +deal> ll ? tal; directly
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An introduction to Systems of Systems Control
*onsensus2!ased *ontrol
Step% > o!jecti#e
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An introduction to Systems of Systems Control
*onsensus2!ased *ontrol
Step9 > *oordination aria!les
(Mo Aamshidi)
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An introduction to Systems of Systems Control
*onsensus2!ased *ontrol
Step? > *entrali1ed strategy
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An introduction to Systems of Systems Control
*onsensus2!ased *ontrol
StepB > *onsensus 8uilding
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An introduction to Systems of Systems Control
5et-or;ed control
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One of the main challenges in 5*SCs is the loss ordelays in transmission and receipt of data from sensors to
controllers and from controllers to actuators.
he challenge in SoS net-or;ed control is to de#elop an SoS
distri!uted control system -hich can tolerate lost pac;ets$
partially decoded pac;ets$ delays$ and fairness issues i.e.add ro!ustness to the control paradigm.
An introduction to Systems of Systems Control
5et-or;ed control
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An introduction to Systems of Systems Control
0ierarchical control
A system is chosen to play the role of a coordinator ach system receives its &-' neighbors data via wireless communication
An optimal controller is designed for each system
The coordination of the & solutions via an iterative algorithm gives an
optimal solution of the SoS
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0ierarchical control
10 Model&Coordination Method
*onsider the follo-ing optimi1ation pro!lem>
Minimi1e , ,Su!ject to , , = 0.
-here is the state #ector$ control #aria!les$ is a #ector of interaction !et-eensu!systems$ ∙ an o!jecti#e function$ and (∙) a constraint function./et the pro!lem and its o!jecti#e function !e decomposed into t-o su!systems>
, , = , , + (, , )and , , , = 0, = 1,2
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0ierarchical control
10 Model&Coordination Method
2irst&level "roblem subsystem Eind
= min , (
,
,
)su!ject to ( , , , ) = 0Second&level "roblemmin
= + ()
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0ierarchical control
30 -oal&Coordination Method
*onsider the same optimi1ation pro!lem> Minimi1e , , Su!ject to , , = 0.+n this method$ the interaction is completely remo#ed !y cutting all the lin;s among the
su!systems . /et !e the outgoing #aria!le from the th su!system$ and thecorresponding input. Since the interaction is remo#ed$ itCs o!#ious that ≠ .he glo!al pro!lem is completely decoupledF the su!systems pro!lems are completely
independent. he ne- formulation of the o!jecti#e functions is>
, , , = 0 , , , = 0+t is necessary that the interaction balance principle !e satisfied ( = ).*onsider the follo-ing ne- cost function>
, , , , = , , + , , + ( − )Ghere is a #ector of -eighting parameters -hich causes any interactionun!alance − to affect the o!jecti#e function.
− =
− +
− .
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2irst level "roblem.
Subsystem 1.
min,!,",#$ , , , +
+
su!ject to , , , = 0Subsystem 3.
min$
,!$
,"$
,#
, , , + + su!ject to , , , = 0
0ierarchical control
30 -oal&Coordination Method
Second level "roblem (coordinator).
min% & = min% ( − )
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0ierarchical control
o apply these methods$ t-o principles can !e applied> the +nteraction prediction
principle and the interaction !alance principle.
*onsider the follo-ing large2scale linear time2in#ariant system>
' = + * , = Ghere -. are - / 1 -. / 1 state and control #ectors. +t is assumed that (thea!o#e system) can !e decomposed into
' = + * + , 0 = , = 1,2, , 3Ghere the interaction #ector
= 4 566 7
6869is a linear com!ination of the states of the other 3 − 1 su!systems$ and 56 is an - / -6matriH.
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0ierarchical control
$%am"le
*onsider the follo-ing %9th2order system
' =
0 1 0 0 0 0 0 0 0 0 0 00 0 1 0 0 0 0 0 0 0 0 0−: −2 −1 0 0 0 0 0 0 1 0 00 0 0 0 1 0 0 0 0 0 0 00 0 0 0 0 1 0 0 0 0 0 01 0 0 −1 −: −2 0 1 0 0 0 00 0 0 0 0 0 0 1 0 0 0 00 0 0 0 0 0 0 0 1 0 0 00 0 0 0 1 0 −1 −2 −: 0 0 00 0 0 0 0 0 0 0 0 0 1 00 0 0 0 0 0 0 0 0 0 0 10 1 0 0 0 0 1 0 0 −: −2 −1
+
0 00 01 00 00 00 00 00 00 10 00 00 0
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his system can !e decomposed into B ?rd2order su!systems -ith state e=uations
' = 0 1 0
0 0 1
− : − 2 − 1
+ 0 00 0
1 0
; ' = 0 1 0
0 0 1
− 1 − : − 2
+ 0 00 0
0 0
,? = 1,2,:,>, ≠ ? gi#en !y>5 = 5
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'ierarchical Control
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'ierarchical Control
nteraction "rediction "rinci"le in Model coordination
Algorithm 1: +nteraction prediction method for *ontinuous2ime Systems.
Ste" 1. Sol#e 5 independent differential matriH ,iccati e=uations>
' = − − + @ − A B C = AStore ; = 1,2, D D , 3 -. 0 E E C.Ste" 3. +nitiali1e an ar!itrary #alue for () and find the corresponding #alue for ()$then sol#e 5 adjoint differential e=uations>
F' = − − @
F − + 4 56
6
7
6869 B F(C) = 0Store F ; = 1,2, D D , 3 -. 0 E E C.
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Ste" 4. Sol#e ' = − @ − @F + () B 0 = Store ; = 1,2, D D , 3 -. 0 E E C.Ste" 5. t the second2le#el (coordinator)$ use the results of Steps 9 and ? to update the
coordinator #ector>
()()GH
=−I()
4 566()7
6869
G
-here J is the num!er of the current iteration.Ste" 6. *hec; for the con#ergence at the second2le#el (coordinator) !y e#aluating the
o#erall interaction error>
& = 4 K − 4 566()76869
− 4 566()76869
.LM
78
NO
if a desired con#ergence is achie#ed$ stop. Other-ise set J = J + 1 and go !ac; to Step 9.
'ierarchical Control
i hi l l
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'i hi l C l
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'i hi l C t l
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'ierarchical Control
Optimal states trajectories
'i hi l C t l
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'ierarchical Control
Optimal states trajectories
'ierarchical Control
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'ierarchical Control