cours sci31_reine talj_a2015 - séances 1 et 2

8/19/2019 Cours SCI31_Reine Talj_A2015 - Séances 1 Et 2

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Modeling, control and observation of

dynamical systems – case of Systems

of Systems (SoS)

Ali Charara

Reine Kfoury [email protected]

[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



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atural disasters management



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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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0ierarchical control


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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


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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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



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*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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*ommunications

0o- -ell can the ro!ots tal; to each other % cannot tal; to ? directly +deal> ll ? tal; directly


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Step% > o!jecti#e


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Step9 > *oordination aria!les

(Mo Aamshidi)


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Step? > *entrali1ed strategy


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StepB > *onsensus 8uilding


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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.


5et-or;ed control


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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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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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10 Model&Coordination Method

2irst&level "roblem subsystem Eind

= min , (

,

,

)su!ject to ( , , , ) = 0Second&level "roblemmin

= + ()


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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


30 -oal&Coordination Method

Second level "roblem (coordinator).

min% & = min% ( − )


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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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$%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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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.


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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Optimal states trajectories

'i hi l C t l


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Optimal states trajectories



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cours sci31_reine talj_a2015 - séances 1 et 2

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