Where

innovationstarts

Event-triggeredControl

Maurice

Heemels

5th

HYCON2PhDSchool

2/6

6

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•Paulo

Tabuada&KalleJohansson(EECIcourse)

•DuarteAntunes

•NiekBorgers

•TijsDonkers

•Tom

Gommans

Acknowledgements

3/6

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Real-timecontrol

•Feedback

controlistypicallyim

plementedonmicroprocessors

usingperi-

odictime-triggeredexecution[1]

mea

sure

x x=

f(x

,u)

com

pute

k(x

) feed

k(x

)

•Mostofexistingcontroltheory

wasdevelopedbyignoringthereal-time

implementationdetails:u(t)=k(x(t))

• Objectofanalysis:x=f(x,k(x))

[1]courtesyPaulo

Tabuada

Introduction

3/6

6

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Real-timecontrol

•Feedback

controlistypicallyim

plementedonmicroprocessors

usingperi-

odictime-triggeredexecution[1]

mea

sure

x x=

f(x

,u)

com

pute

k(x

) feed

k(x

)x

=f(x

,k(x

))=?

•Mostofexistingcontroltheory

wasdevelopedbyignoringthereal-time

implementationdetails:u(t)=k(x(t))

• Objectofanalysis:x=f(x,k(x))

[1]courtesyPaulo

Tabuada

Introduction

3/6

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Real-timecontrol

•Feedback

controlistypicallyim

plementedonmicroprocessors

usingperi-

odictime-triggeredexecution[1]

mea

sure

x x=

f(x

,u)

com

pute

k(x

) feed

k(x

)x

=f(x

,k(x

))=?

•Mostofexistingcontroltheory

wasdevelopedbyignoringthereal-time

implementationdetails.

•Assumption:Computationandcommunicationsufficientlyfast

•Separationofconcernswithperiodictime-triggeredparadigm:

–Controlengineers

designcontrollers

whileignoringim

plementations

–Software

engineers

scheduletaskswhileignoringtheirfunctionality

[1]courtesyPaulo

Tabuada

Introduction

4/6

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PeriodicorAperiodic:That’sthequestion!

•Separationofconcernssimplifiesthedesignprocess,butresults

ininefficientusageofresources

•Paradigm

shift:Periodiccontrol−→

Aperiodiccontrol

•Technologicalmotivation:

Resource-constrainedlarge-scalecyber-physicalsystems

–Computationtimeonembeddedsystems

–Netw

ork

utilisationin

NCSs

–Battery

powerin

WCSs

Introduction

5/6

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Paradigm

shift:Periodiccontrol−→

Aperiodiccontrol

•Event-triggeredcontrol:reactive

u(t)=

K(x(t

k)),whent∈[tk,t

k+1)

t k+1=

inf{t

>t k

|C(x(t),x(t

k))�

0}C

ontr

olle

r

Act

uato

rP

hysi

cal S

yste

mS

enso

r

•Exampleevent-triggeringcondition

‖x(t)−x(t

k)‖

�σ‖x

(t)‖

+δ

Introduction

7/6

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Firsthour

•Basicsetupforevent-triggeredcontrol:State

feedback

case

•Twoanalysisframeworks

–Perturbationapproach

–Hybridsystem

approach

Secondhour

•Output-feedback

event-triggeredcontrol

•Robustness

inevent-triggeredcontrol

•Conclusionsandoutlook

Outline

8/6

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•Linearsystem

x(t)=Ax(t)+Bu(t)

• State

feedback

u(t)=Kx(t)

mea

sure

x x=

f(x

,u)

com

pute

k(x

) feed

k(x

)x

=f(x

,k(x

))=?

•Idealloop:x(t)=(A

+BK)x(t)

• Closed-loopstability:A+BK

Hurw

itzmatrix

Real-timecontrol

9/6

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•Linearsystem

x(t)=Ax(t)+Bu(t)

• State

feedback

u(t)=Kx(t)

mea

sure

x x=

f(x

,u)

com

pute

k(x

) feed

k(x

)x

=f(x

,k(x

))=?

•Idealloop:x(t)=(A

+BK)x(t)

• Sampled-data

controlwithexecutiontimest k,k∈

N

u(t)=Kx(t

k),

t∈[tk,t

k+1)

•Aperiodiccontrol:Inter-executiontimest k

+1−t k

varying

•Perturbationperspective:im

plementation-inducederror

e(t)=x(t

k)−x(t)fort∈[tk,t

k+1)

x(t)=Ax(t)+BKx(t

k)=(A

+BK)x(t)+BKe(t)

[1]Tabuada,TAC2007

Real-timecontrol

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•Linearsystem

x(t)=Ax(t)+Bu(t)

• Sampled-data

controlwithexecutiontimest k,k∈

N

u(t)=Kx(t

k),

t∈[tk,t

k+1)

•Perturbationperspective:

x(t)=

Ax(t)+BKx(t

k)=(A

+BK)x(t)+BKe(t)

• Foridealloop:Since

A+

BK

Hurw

itz,

quadraticLyapunovfunction

V(x)=x� P

xs.t.

V�−c

‖x‖2

•Forperturbedloop:

V�−a

2‖x

‖2+b2‖e‖2

Event-triggeredcontrol

10/6

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•Perturbationperspective:

x(t)=

Ax(t)+BKx(t

k)=(A

+BK)x(t)+BKe(t)

•Since

A+BK

Hurw

itz,quadraticLyapunovfunctionV(x)=x� P

xs.t.

V�−a

2‖x

(t)‖2

+b2‖e(t)‖2

•Question:Howto

designevent-trig.mech.determ

iningt k

s.t.GES?

t k+1=inf{t

>t k

|C(x(t),e(t))�0}

Event-triggeredcontrol

10/6

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•Perturbationperspective:

x(t)=

Ax(t)+BKx(t

k)=(A

+BK)x(t)+BKe(t)

•Since

A+BK

Hurw

itz,quadraticLyapunovfunctionV(x)=x� P

xs.t.

V�−a

2‖x

(t)‖2

+b2‖e(t)‖2

•Question:Howto

designevent-trig.mech.determ

iningt k

s.t.GES?

t k+1=inf{t

>t k

|C(x(t),e(t))�0}

•Crux:Guarantee‖e(t)‖

�σa/b

·‖x(t)‖

with0<

σ<

1s.t.

V�−a

2‖x

(t)‖2

+b2‖e(t)‖2

�−(

1−σ2)a

2‖x

(t)‖2

•GuaranteeforGlobalExponentialStability

t k+1=inf{t

>t k

|‖x(t

k)−x(t)

︸︷︷

︸=e(t)

‖�σa/b

·‖x(t)‖}

Event-triggeredcontrol

11/6

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•Summary

ofevent-triggeredsetup:

–Linearsystem

x(t)=Ax(t)+Bu(t)

–Executiontimest k,k∈

N

t k+1=inf{t

>t k

|‖x(t

k)−x(t)‖

�σa/b

·‖x(t)‖}

–Controllaw:

u(t)=Kx(t

k),

t∈[tk,t

k+1)

•Globalexponentialstability(GES)

•Question:Whichim

portantissueshould

westillverify?

Event-triggeredcontrol

12/6

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•Summary

ofevent-triggeredsetup:

–Linearsystem

x(t)=Ax(t)+Bu(t)

–Executiontimest k,k∈

N

t k+1=inf{t

>t k

|‖x(t

k)−x(t)‖

�σa/b

·‖x(t)‖}

–Controllaw:

u(t)=Kx(t

k),

t∈[tk,t

k+1)

•Minim

alinter-eventtime(M

IET)

inf{t

k+1−t k

|k∈

N}

•MIETshould

have

astrictlypositive

lowerbound!

Minim

alinter-eventtimes

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•Inter-executiontimes:

t k+1=inf{t

>t k

|‖x(t

k)−x(t)

︸︷︷

︸=e(t)

‖�σa/b

·‖x(t)‖}

• Note

afterevent-time:e(t k)=0

• Dynamicsstate:

x(t)=

(A+BK)x(t)+BKe(t)

• Dynamicsinducederrore(t)=x(t

k)−x(t)fort∈[tk,t

k+1)

e(t)

=−x

(t)=−(

A+BK)x(t)−BKe(t)

• Inter-eventtime:timeneededfor‖e(t)‖2

−σ2a2/b

2·‖x(t)‖2

toreach

0

Minim

alinter-eventtimes

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•Note

afterevent-time:e(t k)=0

x(t)=

(A+BK)x(t)+BKe(t)

e(t)

=−x

(t)=−(

A+BK)x(t)−BKe(t)

• Inter-eventtime:timeneededfor‖e(t)‖2

−σ2a2/b

2·‖x(t)‖2

toreach

0

⇓•T

akeξ=(x,e)withdynamics

ξ(t)=Φξ(t)withΦ=

( A+BK

BK

−(A+BK)−B

K

) andξ(0)

=

( I 0) x0

•Inter-eventdeterm

inedbytimeneededforξ�(t)Q

ξ(t)to

reach

0with

Q=

( −σ2a2/b

2I

00

I

)

−→MIET:

inf x

0�=0

inf{t

>0|x

� 0(I

0)eΦ

�t Q

eΦt

( I 0) x0=0}

Minim

alinter-eventtimes

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−→MIET

inf x

0�=0

inf{t

>0|x

� 0(I

0)eΦ

�t Q

eΦt

( I 0) x0=0}

• Thisleadsto

[eliminatingx0]

inf{t

>0|(I0)eΦ

�t Q

eΦt

( I 0) �0}

• Proofofstrictlypositive

lowerboundonMIET(global)

•Computablebynon-conservative

(!)eigenvaluetest

Minim

alinter-eventtimes

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•Linearsystems

x(t)=Ax(t)+Bu(t)

• Sampled-data

controlwithexecutiontimest k,k∈

N

u(t)=Kx(t

k),

t∈[tk,t

k+1)

• Perturbationperspective:

x(t)=

Ax(t)+BKx(t

k)=(A

+BK)x(t)+BKe(t)

•Since

A+BK

Hurw

itz,quadraticLyapunovfunctionV(x)=x� P

xs.t.

V�−a

2‖x

(t)‖2

+b2‖e(t)‖2

•Event-triggeringmechanism

t k+1=inf{t

>t k

|‖x(t

k)−x(t)

︸︷︷

︸=e(t)

‖�σa/b

·‖x(t)‖}

•Guarantees:

–Closed-loopstability

–(Global)lowerboundonminim

alinter-eventtimes

Summary

16/6

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Example

1:State

feedback

control

•Considerx=[ 0

1−2

3] x+[ 0 1] u

andu(t)=[1

−4]x(t

k)

• Exampletakenfrom

[1]

•TTC

:t k

=k·0.025

• ETC

:t k

=t⇐⇒

‖e(t)‖

=0.05‖x

(t)‖

[1]Tabuada,TAC’07

IllustrativeExamples

16/6

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Example

1:State

feedback

control

•Considerx=[ 0

1−2

3] x+[ 0 1] u

andu(t)=[1

−4]x(t

k)

• Exampletakenfrom

[1]

•TTC

:t k

=k·0.025

• ETC

:t k

=t⇐⇒

‖e(t)‖

=0.05‖x

(t)‖

MIET=0.025

02

46

810

12

14

0

0.51

1.5

timet

‖x(t)‖

TTC

ETC

02

46

810

1214

10−4

10−3

10−2

10−1

100

timet

inter-eventtimeτi

TTC

ETC

IllustrativeExamples

16/6

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Example

1:State

feedback

control

•Considerx=[ 0

1−2

3] x+[ 0 1] u

andu(t)=[1

−4]x(t

k)

• Exampletakenfrom

[1]

•TTC

:t k

=k·0.025

• ETC

:t k

=t⇐⇒

‖e(t)‖

=0.05‖x

(t)‖

MIET=0.025

02

46

810

1214

0

0.51

1.5

timet

‖x(t)‖

TTC

ETC

02

46

810

12

14

0

200

400

600

timet

numberofevents

TTC

ETC

IllustrativeExamples

17/6

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•Analysisin

perturbationapproach

basedon

–x(t)=Ax(t)+BKx(t

k)=(A

+BK)x(t)+BKe(t)

–‖e(t)‖

�σa/b

·‖x(t)‖

•Ignoresdynamicserrore(t)=x(t

k)−x(t)fort∈[tk,t

k+1)

e(t)

=−x

(t)=−(

A+BK)x(t)−BKe(t)

• Hybridsystem:ξ=(x,e)

ξ=Φξ

whenξ�Qξ�0

ξ+=Jξ

whenξ�Qξ�0

Φ=

( A+BK

BK

−(A+BK)−B

K

)Q

=

( −σ2a2/b

2I

00

I

)J=

( I0

00)

[1]Donkers,Heemels,CDC10&TAC12

[2]Postoyanetal,CDC11

HybridSystemsApproach

18/6

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•Hybridsystem:ξ=(x,e) ξ=Φξ

whenξ�Qξ�0

ξ+=Jξ

whenξ�Qξ�0

•Stabilityanalysisusinghybridtools[1]:V(ξ)=ξ�Pξ

–d dtV

(ξ)<

0whenξ�Qξ�0

–V(Jξ)

�V(ξ)whenξ�Qξ�0

• Linearmatrixinequalities:ifthere

areα,β�0s.t.

–Φ

� P+PΦ−αQ

≺0

–J

� PJ−P+βQ

�0

• GuaranteeforGES

•Nevermore

conservative

thanperturbationapproach

[2]

[1]Goebel,Sanfelice,Teel,CSM09

[2]Donkers,Heemels,TAC12

HybridSystemsApproach

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Example

1:ComparisonPandHSapproach

•Considerx=

[ 01

−23] x

+[ 0 1] u

andu(t)=[1

−4]x(t

k)

• Exampletakenfrom

[1]

•Welookforlargestσgivingstability:‖e‖2

�σ‖x

‖2[2]

σMIET

P:Resultsfrom

[1]

0.0030

0.0318

P:Byminim

isingtheL 2

-gain

0.0273

0.0840

HybridSystem

0.0588

0.1136

•PS:viaminim

isingL 2

-gain:minim

iseb/ato

maximisea/b

V�−a

2‖x

(t)‖2

+b2‖e(t)‖2

forx=(A

+BK)x

+BKe

• ETM

:t k

+1=inf{t

>t k

|‖x(t

k)−x(t)

︸︷︷

︸=e(t)

‖�σa/b

·‖x(t)‖}

[1]Tabuada,TAC’07

[2]Donkers,Heemels,CDC10&TAC12

IllustrativeExamples

20/6

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Dosimilarresultshold

forthenonlinearcase?

•Nonlinearsystem

x=f(x,u)

• Sampled-data

controlwithexecutiontimest k,k∈

N

u(t)=k(x(t

k))=k(x(t)+e(t)),

t∈[tk,t

k+1)

•Perturbationperspective:x=f(x,k(x

+e))

•SupposeInput-to-State

Stability(ISS)withISSLF

V

∂V ∂xf(x,k(x

+e))�−α

(‖x‖)

+β(‖e

‖)

•α,βclassK-

functions:α(0)=0,

continuous,strictlyincreasing

•Event-triggeringmechanism:forsome0<

σ<

1

−α(‖x

‖)+β(‖e

‖)�−σ

α(‖x

‖)•t

k+1=inf{t

>t k

|β(‖e

(t)‖)

�−(

1−σ)α(‖x

(t)‖)

}• S

tability&lowerboundonminim

alinter-eventtimes(Lipschitz)

NonlinearSystems

22/6

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Complete

design

•Emulation-baseddesignprocedure

forK

• Usingperturbationapproach

wecansyn

thesize

ETM

‖e‖�

σ‖x

‖• E

TM(σ)canbepushedfurtherusinghybridsystem

approach

•GuaranteedGES

•Guaranteedpositive

andgloballowerboundonMIET

Are

wedone??

Event-triggeredcontrol

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•Whatiffullstate

xnotavailableforfeedback,butonlyoutputy?

Output-basedETC

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Illustrativeexample

•Consider { x

p=[ 1

−110

−1] x

p+[ 1 1] u

y=

[10]x

p

u(t)=−2

y(t

k)

•ETM

:‖y

(t)−y(t

k)‖2

>σ‖y

(t)‖2

•Parameter:σ=0.5

Output-basedETC

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Illustrativeexample

•Minim

alinter-eventtime(M

IET)iszero!

•Zenobehavior:

aninfinitenumberofevents

inafinitelength

in-

terval

Output-basedETC

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•Whatifsmalldisturbancesare

present?

Disturbancesin

ETC

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Illustrativeexample

•Considerx=[ 0

1−2

3] x+[ 0 1] u

andu(t)=[1

−4]x(t

k)

• TTC

:t k

=k·0.025

• ETC

:t k

=t⇐⇒

‖e(t)‖

=0.05‖x

(t)‖

02

46

810

1214

0

0.51

1.5

timet

‖x(t)‖

TTC

ETC

02

46

810

1214

10−4

10−3

10−2

10−1

100

timet

inter-eventtimeτi

TTC

ETC

Disturbancesin

ETC

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Illustrativeexample

•Considerx=[ 0

1−2

3] x+[ 0 1] u

+wandu(t)=[1

−4]x(t

k)

• TTC

:t k

=k·0.025

• ETC

:t k

=t⇐⇒

‖e(t)‖

=0.05‖x

(t)‖

02

46

810

12

14

0

0.51

1.5

timet

‖x(t)‖

TTC

ETC

02

46

810

1214

10−4

10−3

10−2

10−1

100

timet

inter-eventtimeτi

TTC

ETC

Disturbancesin

ETC

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Issuesin

ETC

•Output-basedETC

•Robustness

ofMIET/resourceutilization

Outline2ndhour

•SolutionIforoutput-basedETC

:

–Alternative

event-triggeringconditions(notrelative)

•Event-separationpropertiesforalternative

ETC

(robustness)

•SolutionIIforoutput-basedETC

:

–Timeregularization:PeriodicEvent-TriggeredControl(PETC

)

•Again

perturbationandhybridsystem

approaches

•Conclusions&Outlook

What’snext?

30/6

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Output-basedcontrol

31/6

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Possible

remedies

•SolutionI:Adoptalternative

ETM

sinsteadof‖y

−y‖2

>σ‖y

‖2–Absolute:‖y

−y‖2

>ε

–Mixed:‖y

−y‖2

>σ‖y

‖2+ε

•SolutionII:Timeregularization

–Enforceminim

alinter-eventtimeT[1,2]

t k+1=inf{t

>t k+T|‖

y−

y‖2

>σ‖y

‖2}

–Events

onlyatkh,k∈N

t k+1=inf{t

>t k

|‖y−

y‖2

>σ‖y

‖2∧t=kh,k∈N}

�PeriodicEvent-TriggeredControl(PETC

)

[1]Tallapragada&Chopra.CDC12&NECSYS12

[2]Heemels,Sandee,VandenBosch,IJC08

Output-basedETC

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Output-basedETC

usingMixedETM

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System

Description

Objective:

•Setupanoutput-basedevent-triggeringmechanism

(ETM

)

•GuaranteedMIET>

0

• MixedETM

:‖y

i−

yi ‖

>σi‖y

i ‖+ε i

•Generalsetup:decentralizedETM

[Donkers,Heemels,TAC2012]

TheEvent-TriggeredControlSystem

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•MixedETM

:‖y

i−

yi ‖

>σi‖y

i ‖+ε i

•Minim

alinter-eventtimestrictlypositive

guaranteed(boundde-

pendson‖x

(0)‖

and‖w

‖)•U

ltim

ate

boundedness:determ

inedbyσi

•Sizeultim

ate

bound:tunedbyε i

•TwoperspectivesforLM

I-basedstabilityandL ∞

-gain

analysis

–Hybridsystem

approach

–Perturbationapproach

–HSless

conservative,butharderwork

•Tradeoffultim

ate

bound/L

∞-gain

andnumberofevents

[Donkers,Heemels,TAC2012]

OverviewformixedETMs

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Example

2:ε i=0zero

inter-eventtimes!

•Consider { x

p=[ 0

1−2

3] xp+[ 0 1] u

y=

[−1

4]x

p

{ xc=[ 0

10

−5] x

c+[ 0 1] y

u=

[1−4

]xc

•ETM

:‖y

−y‖2

=σ1‖y

‖2+ε 1

and‖u

−u‖2

=σ1‖u

‖2+ε 2

•Parameters:σ1=σ2=10

−3andε 1

=ε 2

=0

IllustrativeExamples

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Example

2:ε i=0zero

inter-eventtimes!

•Consider { x

p=[ 0

1−2

3] xp+[ 0 1] u

y=

[−1

4]x

p

{ xc=[ 0

10

−5] x

c+[ 0 1] y

u=

[1−4

]xc

•ETM

:‖y

−y‖2

=σ1‖y

‖2+ε 1

and‖u

−u‖2

=σ1‖u

‖2+ε 2

•Parameters:σ1=σ2=10

−3andε 1

=ε 2

=0

00.05

0.1

0.15

0.2

-150

-100-50050

100

u,nodei=

2y,nodei=

1

timet

00.05

0.1

0.15

0.2

10−1

5

10−1

0

10−5

nodei=

2nodei=

1

timet

IllustrativeExamples

35/6

6

/w

Example

2:NeedforextendingETM

includingε i

•Consider { x

p=[ 0

1−2

3] xp+[ 0 1] u

y=

[−1

4]x

p

{ xc=[ 0

10

−5] x

c+[ 0 1] y

u=

[1−4

]xc

•ETM

:‖y

−y‖2

=σ1‖y

‖2+ε 1

and‖u

−u‖2

=σ1‖u

‖2+ε 2

•Parameters:σ1=σ2=10

−3andε 1

=ε 2

=10

−3

05

1015

2025

30

-60

-40

-200204060

t imet

05

1015

2025

30

10−4

10−3

10−2

10−1100

nodei=

2nodei=

1

timet

IllustrativeExamples

36/6

6

/w

Example

3:WhatETC

isallabout!

•Consider

{ xp=[ 0

1−2

−3] x

p+[ 0 1] u

+[ 0 1] w

y=

[1

0]x

p

{ xc=[ −2

1−1

3−3] x

c+[ −2 −5

] y

u=

[5

2]x

c

•Takingσ1=

σ2=

10−3

andε 1

=ε 2

=10

−3,yieldsL ∞

-gain

of0.46

forz=[10]xp

05

1015

2025

3010

−2

10−1100

101

nodei=

2nodei=

1

timet

•Actwhenneeded!

IllustrativeExamples

37/6

6

/w

ETC

underdisturbances

40/6

6

/w

Event-separationproperties

•Considerx=Ax+Bu+wandu(t)=Kx(t

k)=K(x(t)+e(t))

•Inter-executiontimes:

t k+1=inf{t

>t k

|‖x(t

k)−x(t)

︸︷︷

︸=e(t)

‖�σ‖x

(t)‖

+ε}

→MIETτ(x

0,w

)dependentonx0andw:

τ(x

0,w

)=inf k

∈N(t

k+1−t k)

→Differentkindsoftriggeringconditions

•Relative

ETM

:ε=0andσ>

0

• Absolute

ETM

:ε>

0andσ=0

• MixedETM

:ε>

0andσ>

0

Disturbancesin

ETC

/w

Event-separationproperties

Rnx

τ(x

0,0)�

τ min

Lowerboundτ m

in>

0Global:

forthecomplete

Rnx

inabsence

ofdisturbances

Rnx

τ(x

0,0)�

τ min

XLowerboundτ m

in>

0Semi-global:

foranycompactsetX

⊂Rnx

inabsence

ofdisturbances

Rnx

τ(x

0,0)�

τ min

x0

Lowerboundτ m

in>

0Local:

foranysinglepointx0∈Rnx

inabsence

ofdisturbances

/w

Event-separationproperties

Rnx τ(x

0,w

)�

τ min

Robust

Lowerboundτ m

in>

0Global:

forthecomplete

Rnx

for

bounded

disturbances

‖w‖ L

∞�

ε

Rnx τ(x

0,w

)�

τ min

XRobust

Lowerboundτ m

in>

0Semi-global:

foranycompactsetX

⊂Rnx

for

bounded

disturbances

‖w‖ L

∞�

ε

Rnx τ(x

0,w

)�

τ min

x0

Robust

Lowerboundτ m

in>

0Local:

foranysinglepointx0∈Rnx

for

bounded

disturbances

‖w‖ L

∞�

ε

/w

Overviewofevent-separationproperties

Robust

global

Robust

semi-global

Robust

local

Global

Semi-global

Local

�

/w

Overviewofevent-separationproperties

Robust

global

Robust

semi-global

Robust

local

Global

Semi-global

Local

� � �

� � �

/w

Overviewofevent-separationproperties

Robust

global

Robust

semi-global

Robust

local

Global

Semi-global

Local

×

/w

Overviewofevent-separationproperties

Robust

global

Robust

semi-global

Robust

local

Global

Semi-global

Local

××

××

××

/w

Example:Linearsystem

P

ETM

C

ZOH

ua=

u

x

xc

P:x=

[0

1−2

3

] x+

[ 0 1

] u

C:u=

[ 1−4

] (x+e)

relETM

t i=

t⇐⇒

‖e(t)‖

=0.05‖x

(t)‖

absETM

t i=

t⇐⇒

‖e(t)‖

=0.001

mixETM

t i=

t⇐⇒

‖e(t)‖

=0.05‖x

(t)‖

+0.001

/w

Example:Linearsystem

P

ETM

C

ZOH

ua=

u

x

xc

P:x=

[0

1−2

3

] x+

[ 0 1

] u

C:u=

[ 1−4

] (x+e)

02

46

810

1214

0

0.51

1.5

timet

‖x(t)‖

relETM

02

46

810

1214

10−4

10−3

10−2

10−1

100

timet

inter-eventtimeτi

relETM

/w

Example:Linearsystem

ETM

robustglobal

global

robustsemi-global

semi-global

robustlocal

local

relative

��

�absolute

�

�mixed

��

�

02

46

810

12

14

0

0.51

1.5

timet

‖x(t)‖

relETM

absETM

mix

ETM

02

46

810

12

14

10−

4

10−

3

10−

2

10−

1

100

timet

inter-eventtimeτi

relETM

absETM

mix

ETM

/w

Example:Linearsystem

P

ETM

C

ZOH

ua=

uw

x

xc

P:x=

[0

1−2

3

] x+

[ 0 1

] u+

w

C:u=

[ 1−4

] (x+e)

relETM

t i=

t⇐⇒

‖e(t)‖

=0.05‖x

(t)‖

absETM

t i=

t⇐⇒

‖e(t)‖

=0.001

mixETM

t i=

t⇐⇒

‖e(t)‖

=0.05‖x

(t)‖

+0.001

/w

Example:Linearsystem

P

ETM

C

ZOH

ua=

uw

x

xc

P:x=

[0

1−2

3

] x+

[ 0 1

] u+

w

C:u=

[ 1−4

] (x+e)

02

46

810

12

14

0

0.51

1.5

timet

‖x(t)‖

relETM

02

46

810

1214

10−4

10−3

10−2

10−1

100

timet

inter-eventtimeτi

relETM

/w

Example:Linearsystem

ETM

robustglobal

global

robustsemi-global

semi-global

robustlocal

local

relative

�

�

�

absolute

××

��

��

mixed

��

��

��

02

46

810

1214

0

0.51

1.5

timet

‖x(t)‖

relETM

absETM

mix

ETM

02

46

810

1214

10−4

10−3

10−2

10−1

100

timet

inter-eventtimeτi

relETM

absETM

mix

ETM

41/6

6

/w

•Event-separation

properties:

local/semi-global/globaland

ro-

bustness

ofMIET/resourceutilization

State-feedback

case

ETM

robustglobal

global

robustsemi-global

semi-global

robustlocal

local

relative

�

�

�

absolute

××

��

��

mixed

��

��

��

Output-feedback

case

ETM

robustglobal

global

robustsemi-global

semi-global

robustlocal

local

relative

××

××

××

absolute

××

��

��

mixed

××

��

��

•Relative

triggeringfragile.Zero

robustness

•Mixedorabsolute

effective(semi-global)butUB(noGAS)

[Borgers

&Heemels,CDC13submitted]

Summary

42/6

6

/w

PeriodicEvent-TriggeredControl(PETC

)

43/6

6

/w

•SolutionI:Adoptalternative

ETM

sinsteadof‖y

−y‖2

>σ‖y

‖2–Mixed:‖y

−y‖>

σ‖y

‖+ε

•SolutionII:Timeregularization

–Events

onlyatkh,k∈N

t k+1=inf{t

>t k

|‖y−

y‖2

>σ‖y

‖2∧t=kh,k∈N}

�PeriodicEvent-TriggeredControl(PETC

)[1,2]

[1]Heemels,Donkers

&Teel,CDC/ECC’11

[2]Heemels,Donkers

&Teel,TAC13

PeriodicETC

44/6

6

/w

•Paradigm

shift:Periodiccontrol−→

Event-triggeredcontrol

•IncontrastwithCETC

....

•....Samplingperiodic,but...

•....Every

samplingtimeitisdecidedto

actornot.

•PETC

vs.CETC

:Im

plementationadvantages!

–Guaranteed(reasonable)minim

alinter-eventtime

–Onlytime-periodicverificationofevent-triggeringconditions

–More

inlinewithtime-slicedarchitectures

PeriodicETC

(PETC

)45/6

6

/w

Description

d dtx

=A

px+B

pu+B

ww

u(t)=Kx(t)

x(t)=

{ x(t

k),

whenC(x(t

k),x(t

k))>

0

x(t

k),

whenC(x(t

k),x(t

k))�

0fort∈(t

k,t

k+1]

witht k

=khandh>

0fixedsamplingperiod

PETC

46/6

6

/w

u(t)=Kx(t),

x(t)=

{ x(t

k),

whenC(x(t

k),x(t

k))>

0

x(t

k),

whenC(x(t

k),x(t

k))�

0

• [1,2]

‖x(t

k)−x(t

k)‖

>σ‖x

(tk)‖

• [3]

‖Kx(t

k)−Kx(t

k)‖

>σ‖K

x(t

k)‖

• [4,5]

(Ax(t

k)+BKx(t

k))

� P(A

x(t

k)+BKx(t

k))>

βx� (t k)P

x(t

k)

C(ξ(t k))=ξ�(t

k)Q

ξ(t k)>

0,

where

ξ:=

(x,x)∈R

nξ

[1]Tabuada,TAC’07

[3]Donkers

&HeemelsCDC’10,TAC’12

[2]Wang&Lemmon,TAC’10

[4]Velasco

etal,CDC’09

[5]Mazo

etal,ECC’09

PETC

47/6

6

/w

Hybridsystem

form

ulation

d dt[ ξ τ

] =

[ Aξ+Bw

1

] ,whenτ∈[0,h

],

[ ξ+ τ+

] =

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩[ J1ξ 0

] ,whenξ�

Qξ>

0,τ=h

[ J2ξ 0

] ,whenξ�

Qξ�

0,τ=h

withξ:=

(x,x)∈R

nξand

A:=

[ ApB

pK

00

] ,B

:=

[ Bw 0

] ,J1:=

[ I0

I0] ,

J2:=

[ I0

0I

]

PETC

48/6

6

/w

d dt[ ξ τ

] =

[ Aξ+Bw

1

] ,whenτ∈[0,h

],

[ ξ+ τ+

] =

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩[ J1ξ 0

] ,whenξ�

Qξ>

0,τ=h

[ J2ξ 0

] ,whenξ�

Qξ�

0,τ=h

Problem

form

ulation:

•Globallyexponentiallystable

(w=0):

‖ξ(t)‖

�ce

−ρt ‖ξ

(0)‖

•L2gain

smallerthanorequalto

γwithz=Cξ+Dw

√ ∫∞

0

‖z(t)‖2

dt�

γ

√ ∫∞

0

‖w(t)‖2

dt

PETC

49/6

6

/w

Problem

form

ulation:

•Globallyexponentiallystable

‖ξ(t)‖

�ce

−ρt ‖ξ

(0)‖

•L2gain

smallerthanorequalto

γwithz=Cξ+Dw

√ ∫∞

0

‖z(t)‖2

dt�

γ

√ ∫∞

0

‖w(t)‖2

dt

Threeapproaches:

(i)Discrete-tim

epiecewiselinearsystem

(PWL)approach

(ii)Discrete-tim

ePerturbationapproach

→Sim

ilar

(iii)Hybridsystem

approach

[1]Heemels,Donkers

&Teel,TAC13

PETC

50/6

6

/w

d dt[ ξ τ

] =

[ Aξ+Bw

1

] ,whenτ∈[0,h

],

[ ξ+ τ+

] =

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩[ J1ξ 0

] ,whenξ�

Qξ>

0,τ=h

[ J2ξ 0

] ,whenξ�

Qξ�

0,τ=h

• Discretize

att k

=khwithξ k

:=ξ(t k)(w

=0)

ξ k+1=

{ A1ξ k,

whenξ� k

Qξ k

>0,

A2ξ k,

whenξ� k

Qξ k

�0,

where

A1

=eA

hJ1=[ A+

BK

0I

0] ,A

2=eA

hJ2=[ AB

K0

I

] ,

A:=

eAph,

B:=

∫ h 0

eApsB

pds.

Piecewiselinearsystem

approach

51/6

6

/w

ξ k+1=

{ A1ξ k,

whenξ� k

Qξ k

>0,

A2ξ k,

whenξ� k

Qξ k

�0,

where

•PiecewisequadraticLyapunovfunction:

V(ξ)=

{ ξ�P1ξ,

whenξ�

Qξ>

0,

ξ�P2ξ,

whenξ�

Qξ�

0,

Theorem

ThePETC

system

isGES,ifthere

are

matricesP1,P2and

scalars

αij�

0,βij�

0andκi�

0,s.t.

Pi−A

� iPjA

i+(−

1)i α

ijQ+(−

1)jβijA

� iQA

i�

0,foralli,j∈{1,2},

and

Pi+(−

1)i κ

iQ�

0,foralli∈{1,2}.

Piecewiselinearsystem

approach

52/6

6

/w

d dt[ ξ τ

] =

[ Aξ+Bw

1

] ,whenτ∈[0,h

],

[ ξ+ τ+

] =

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩[ J1ξ 0

] ,whenξ�

Qξ>

0,τ=h

[ J2ξ 0

] ,whenξ�

Qξ�

0,τ=h

Main

ideaL 2

gain

analysis:z=Cξ+Dw

•Tim

er-dependentquadraticLyapunovfunctionV(ξ,τ)=ξ�

P(τ)ξ

•d dtV

�−2

ρV

−γ−2z�

z+w

� w

• Duringjumpsdecrease:

V(J

1ξ,0)

�V(ξ,h

),forallξwithξ�

Qξ>

0,

V(J

2ξ,0)

�V(ξ,h

),forallξwithξ�

Qξ�

0

Hybridsystem

approach

52/6

6

/w

Main

ideaL 2

gain

analysisV(ξ,τ)=ξ�

P(τ)ξ

•d dtV

�−2

ρV

−γ−2z�

z+w

� w

• Duringjumpsdecrease:

V(J

1ξ,0)

�V(ξ,h

),forallξwithξ�

Qξ>

0,

Niceidea...buthow?

Hybridsystem

approach

53/6

6

/w

Main

ideaL 2

gain

analysisV(ξ,τ)=ξ�

P(τ)ξ

•d dtV

�−2

ρV

−γ−2z�

z+w

� w

• Duringjumpsdecrease:

V(J

1ξ,0)

�V(ξ,h

),forallξwithξ�

Qξ>

0,

Niceidea...buthow?

•Riccatidifferentialequation(M

=(γ

−2D

� D−I)−

1)

d dτP

=−A

� P−P

A−2

ρP−γ

−2C

� C−(

PB+γ−2C

� D)M

(B� P

+γ−2D

� C)

• HamiltonianH

:=[ A+

ρI+γ−2

BM

D� C

BM

B�

−C� L

C−(

A+ρI+γ−2

BM

D� C

)�]

•F(τ):=

e−Hτ=[ F 11(

τ)

F12(τ)

F21(τ)

F22(τ)]

•P0=(F

21(h)+F22(h)P

h)( F

11(h)+F12(h)P

h

) −1

Hybridsystem

approach

54/6

6

/w

Theorem:Supposethatthere

existmatrixPh�

0,andscalars

μi�

0,

such

thatfori∈{1,2}

⎡ ⎣Ph+(−

1)i μ

iQJ� iF

−� 11PhS

J� i(F

−� 11PhF

−1 11+F21F

−1 11)

I−S� P

hS

0

F−� 11

PhF

−1 11+F21F

−1 11

⎤ ⎦ �0

Then,thePETC

system

isGESwithconvergence

rate

ρ(whenw

=0)

andhasanL 2

-gain

smallerthanorequalto

γ.

Hybridsystem

approach

55/6

6

/w

(i)Piecewiselinearsystem

approach

(ii)Perturbationapproach

(discrete-tim

elinearsystem)

(iii)Hybridsystem

approach

(intersamplebehavior)

•AllLM

I-basedandthusefficient

•(ii):Sim

ple:H ∞

-norm

calculationsforGES

•(i):Leastconservative

forGES

•(iii):L 2

gain

analysis

[1]Heemels,Donkers

&Teel,TAC13

Comparisonoftheapproaches

56/6

6

/w

d dtx

=

[ 01

−23] x

+

[ 0 1] u

u(t)=Kx(t)

K=[1

−4]andt k

=khwithh=0.05.Fort∈(t

k,t

k+1]

x(t)=

{ x(t

k),

when‖x

(tk)−x(t

k)‖

>σ‖x

(tk)‖

x(t

k),

when‖x

(tk)−x(t

k)‖

�σ‖x

(tk)‖

Numericalexample

57/6

6

/w

x(t)=

{ x(t

k),

when‖x

(tk)−x(t

k)‖

>σ‖x

(tk)‖

x(t

k),

when‖x

(tk)−x(t

k)‖

�σ‖x

(tk)‖

MaximalσforwhichGESguaranteedbydifferentapproaches:

(i)σpert:=

0.1728

(H∞

=1/0.1728)

(ii)σPWL=σHS=0.2425

Asexpected,σpert�

σPWLandσHS�

σPWL

Minim

alinter-eventtimes:

•2h=0.1in

form

ercasesand3h

=0.15

inlattercase!

Numericalexample

57/6

6

/w

d dtx

=

[ 01

−23] x

+

[ 0 1] u

u(t)=Kx(t)

K=[1

−4]andt k

=khwithh=0.05.Fort∈(t

k,t

k+1]

x(t)=

{ x(t

k),

when‖K

x(t

k)−Kx(t

k)‖

>σ‖K

x(t

k)‖

x(t

k),

when‖K

x(t

k)−Kx(t

k)‖

�σ‖K

x(t

k)‖

•Note:Could

alsobeoutput-basedexample

u=yandy=Kx

Numericalexample

58/6

6

/w

d dtx

=

[ 01

−23] x

+

[ 0 1] u

u(t)=Kx(t)

K=[1

−4]andt k

=khwithh=0.05.Fort∈(t

k,t

k+1]

x(t)=

{ x(t

k),

when‖K

x(t

k)−Kx(t

k)‖

>σ‖K

x(t

k)‖

x(t

k),

when‖K

x(t

k)−Kx(t

k)‖

�σ‖K

x(t

k)‖

MaximalσforwhichGESguaranteedbydifferentapproaches:

(i)σPWL=0.2550

(ii)σpert=0.2506

(iii)σHS=0.2532

Asexpected,σpert�

σPWLandσHS�

σPWL

Numericalexample

59/6

6

/w

d dtx

=

[ 01

−23] x

+

[ 0 1] u+

[ 1 0] w,

u(t)=Kx(t)

x(t)=

{ x(t

k),

when‖K

x(t

k)−Kx(t

k)‖

>σ‖K

x(t

k)‖

x(t

k),

when‖K

x(t

k)−Kx(t

k)‖

�σ‖K

x(t

k)‖

•L2gain

analysisz=[01]x

00.05

0.1

0.15

0.2

0.25

0246810

γ

σ

•σ↓0

thenγ→

L 2-gain

ofperiodics-d

control(design)

Numericalexample

60/6

6

/w

d dtx

=

[ 01

−23] x

+

[ 0 1] u+

[ 1 0] w,

u(t)=Kx(t)

x(t)=

{ x(t

k),

when‖K

x(t

k)−Kx(t

k)‖

>σ‖K

x(t

k)‖

x(t

k),

when‖K

x(t

k)−Kx(t

k)‖

�σ‖K

x(t

k)‖

•L2gain

analysisz=[01]x

05

1015

2025

30

-1

-0.50

0.51

output

zdisturban

cew

time

t

05

1015

2025

300

0.2

0.4

0.6

0.81

time

t

•σ=0.2

• Inter-eventtimes:0.05

to0.85

(17timesh)!!!!

Numericalexample

61/6

6

/w

•Part

I:Event-triggeredcontrolbasedonfullstate

feedback

andrelative

triggering

–Globalasymptoticstability

–GuaranteedglobalMIET

•Twoanalysisframeworks

–Perturbationperspective(emulation-baseddesign)

–Hybridsystemsperspective(lessconservative)

•Outputfeedback

anddisturbancesremovedfavorablefeatures

•PartII:Output-basedETC

andevent-separationproperties(ESP)

–Absolute/M

ixedtriggering

�UB/p

racticalstability&robust(semi-global)ESP

–Timeregularization(PeriodicEvent-TriggeredControl(PETC

))

�Time-regularizationplusrelative

ETM

:GAS,robustglobalESP,

but

reducesto

time-triggeredloopunderdisturbances

–Carefulwithrobustnessissuesin

resourceutilization

–Again

perturbationandhybridsystem

approaches

Conclusions

62/6

6

/w

•Linearsystem:x=Ax+Bu+w

• Controllaw:u=Kx(t

k)=K(x

+e)

• Executiontimes:

t k+1=inf{t

>t k

|‖x(t

k)−x(t)

︸︷︷

︸=e(t)

‖�σ‖x

(t)‖

+ε}

Extensions&Outlook

62/6

6

/w

•Linearsystem:x=Ax+Bu+w

• Controllaw:u=Kx(t

k)=K(x

+e)

• Executiontimes:

t k+1=inf{t

>t k

|‖x(t

k)−x(t)‖

�σ‖x

(t)‖

+ε}

•Decentralizedevent-triggeredcontrol[1-6]

t k+1=inf{t

>t k

|‖y i(t

k)−y i(t)‖

�σi‖y

i(t)‖+

ε i}

[1]Mazo

Jr.andTabuada,TAC11

[2]Persisetal,Automatica

2013

[3]Garcia,Antsaklis,ACC2012

[4]Wang,Lemmon,TAC11

[5]Donkers,Heemels,TAC2012

[6]Heemels,Donkers,TAC13,Automatica

13

Extensions&Outlook

62/6

6

/w

•Linearsystem:x=Ax+Bu+w

• Controllaw:u=Kx(t

k)=K(x

+e)

• Executiontimes:

t k+1=inf{t

>t k

|‖x(t

k)−x(t)‖

�σ‖x

(t)‖

+ε}

•Model-basedevent-triggeredcontrol[1,2,3,4,5]

–“Iknowwhatyouknowprinciple”

–t k

+1=inf{t

>t k

|‖xpred(t)−x(t)‖

�σ‖x

(t)‖

+ε}

with

xpred(t)=(A

+BK)x

pred(t)withxpred(t

k)=x(t

k)

[1]Yook,Tilbury

etal,TC

ST0

2[2]Lunze,Lehmann,Automatica10

[3]Garcia,Antsaklis,ACC12

[4]Heemels,Donkers,Automatica13

[5]Bernardini,Bemporad,Automatica12

Extensions&Outlook

62/6

6

/w

•Linearsystem:x=Ax+Bu+w

• Controllaw:u=Kx(t

k)=K(x

+e)

• Executiontimes:

t k+1=inf{t

>t k

|‖x(t

k)−x(t)‖

�σ‖x

(t)‖

+ε}

•Outperform

ingperiodictime-triggeredcontrol[1,2,3]

00.

51

1.5

22.

53

3.5

44.

55

01234567

Ave

rage

inte

r−tra

nsm

issi

on ti

me,

hav

g

Average cost

J c J d(h)

β V

(x0)

True

cos

t STC

β =

1.25

[1]Astrom,Bernhardsson,IFAC99,CDC02

[2]Barradas-Berglindetal,NMPC12

[3]Antunes,Heemels,Tabuada,CDC12

Extensions&Outlook

63/6

6

/w

•Event-triggeredandself-triggeredcontrolappealingresearcharea

•System

theory

farfrom

mature

•Manyinterestingproblemsopen

•Goodareaforyoungresearchers!

•Homepage

http://www.dct.tue.nl/heemels

Extensions&Outlook

64/6

6

/w

•Lecturesbasedon:

–D.P.Borgers

andW.P.M

.H.Heemels,Event-separationpropertiesofevent-

triggeredcontrolsystems,IEEEConference

onDecisionandControl2013,

submitted.

–M.C.F.Donkers

andW.P.M

.H.Heemels,Output-BasedEvent-TriggeredCon-

trolwith

Guaranteed

L∞-gain

and

Improved

and

Decentralised

Event-

Triggering,IEEETransactionsonAutomaticControl,57(6),p.1362-1376,

2012.

–W.P.M

.H.Heemels,M.C.F.Donkers,andA.R.Teel,Periodic

Event-Triggered

ControlforLinearSystems,IEEETransactionsonAutomaticControl,58(4),

p.847-861,2013.

–P.

Tabuada,Event-triggered

real-time

scheduling

ofstabilizing

control

tasks,IEEETrans.Autom.Control,vol.

52,no.9,pp.1680-1685,Sep.

2007.

Literature

65/6

6

/w

•Mentionedextensions:

–D.Antunes,W.P.M

.H.Heemels,andPaulo

Tabuada,DynamicProgramming

Form

ulationofPeriodic

Event-TriggeredControl:

Perform

ance

Guarantees

andCo-Design,51stIEEEConference

onDecisionandControl2

012,Hawaii,

USA,p.7212-7217.

–J.D.J.BarradasBerglind,T.M.P.Gommans,andW.P.M

.H.Heemels,Self-

triggeredMPCforconstrainedlinearsystemsandquadraticcosts,IFAC

Conference

onNonlinearModelPredictive

Control2012,Noordwijkerhout,

Netherlands,p.342-348.

–M.C.F.Donkers,P.

Tabuada,andW.P.M

.H.Heemels,"M

inim

um

Attention

ControlforLinearSystems:A

LinearProgrammingApproach",

Discrete

EventDynamic

SystemsTh

eory

andApplications,specialissue"Event-

BasedControlandOptimization,"2012.

–W.HeemelsandM.Donkers,Model-basedperiodicevent-triggeredcontrol

forlinearsystems,Automatica,49(3),March2013,p.698-711,2013.

Literature

66/6

6

/w

•Earlierworks:

–W.P.M

.H.Heemels,J.H.Sandee,P.P.J.vandenBosch,"Analysis

ofevent-

driven

controllers

forlinearsystems",

InternationalJournalofControl,

81(4),pp.571-590(2008)

–W.P.M

.H.Heemels,R.J.A.Gorter,A.vanZijl,P.P.J.v.d.Bosch,S.Weiland,

W.H.A.Hendrix,M.R.Vonder,Asyn

chronousmeasurementandcontrol:

acasestudyonmotorsyn

chronisation,ControlEngineeringPractice,7(12),

1467-1482,(1999)

•Tutorial

–W.P.M

.H.Heemels,K.H.Johansson,andP.

Tabuada,"Anintroductionto

event-triggeredandself-triggeredcontrol,"51stIEEEConference

onDeci-

sionandControl2012,Hawaii,USA,p.3270-3285

•Homepage

http://www.dct.tue.nl/heemels

Literature