A"New"Framework"for"Stability"Analysis" - - PowerPoint PPT Presentation

!! A"New"Framework"for"Stability"Analysis"

• f"Networked"Control"Systems

Oct!16,!2012,!Lund,!Yumiko!Ishido.!!!!!

Research(Interests:!Analysis!and!Synthesis!of!Nonlinear!Systems. Nonlinear!Systems

• Good!tools!for!Linear(Systems.(
• Extensions!for!some(classes(of(

Nonlinear(Systems.((

Extensions Linear!Systems

Goal:!Develop!a!MathemaHcal!Framework!for!Analysis!and! Synthesis!of!Networked!Control!Systems.

(Robust!control!based!on!small!gain! theorem,!IQC!approach,!gain!scheduling,! etc…)

Networked!Control!Systems

Networked"Control"Systems

plant

controller

CommunicaHon! Channel Involving!a!data!rateSlimited! communicaHon!channel!

plant sensor

controller Involving!an!eventS triggered!sensor! plant controller actuator Involving!a!finiteSlevel! valued!actuator

Networked"Control"Systems

plant

controller

CommunicaHon! Channel Involving!a!data!rateSlimited! communicaHon!channel!

plant sensor

controller Involving!an!eventS triggered!sensor! plant controller actuator 3.!Involving!a!finiteS leveled!actuator

FiniteSlevel!quanHzaHon!is!involved!in!the! feedback!loop.!

Classical"Framework"does"not"work?

Ex1:!StabilizaHon!of!an!uncertain!plant!over!a!rateSlimited!communicaHon!channel. bit Unstable!LTI !!!!!Sgain!bounded Small(Gain(Theorem(Is(NOT(Applicable!! Achievable!inputSoutput!property!(MarHns): Unbounded! Growth!Rate Suppose s.t.

Need!for!introducing!a!pracHcal! Local!Stability!Analysis!Framework.

Classical"Framework"does"not"work?

Ex2:!Stability!analysis!of!a!feedback!system!involving!a!uniform!quanHzer. SISO,!LTI uniform!quanHzer S S quanHzaHon!error Stability(Condi@on

A"New"Analysis"Framework"for" Networked"Control"Systems

• 1. Introduce!a!reasonable!noHon!of!local!stability!for!

networked!control!systems.!

• 2. Derive!a!key!theorem!for!stability!analysis.!
• 3. Prepare!a!new!class!of!nonlinearity!that!is!suitable!for!

expressing!quanHzaHon!errors.

holds!for!given!constants!!!!!!!!!!!!!!!!!!!!!!.! A!map!!!!!!is!said!to!be!small!!!!!!signal!!!!!!stable!with!level!!!!!and!input! bound!!!!!if!

Small""""""""Signal""""""""Stability

Small!!!!!!!signal!!!!!!!stability

Local(Boundedness

such!that such!that

" weaker!than!!!!!!!stability!or!finite!gain!!!!!!stability! " equivalent!when!!!!!!!!is!a!linear!map

Small!!!!!!signal!!!!!!!stability!is…

Comparison"with"existing"stabilities

!!!!!!stability Finite!gain!!!!!!stability

such!that such!that

" Defined!with!local!upper!bounds!on!inputSoutput!signals!(not! defined!with!gain).!

Comparison"with"existing"stabilities

Local!!!!!!stability!(Bourles!1996) Small!signal!!!!!!!stability!(Vidyasagar!&!Vanelli,!1982) Small!!!!!!signal!!!!!!!stability!is…

Comparison"with"existing"stabilities

Local!!!!!!!stability! Small!!!!!!!signal!!!!!!stability!

VS “Local(finite(gain(stability” “Local(boundedness”

Classical"Framework"does"not"work?

Ex1:!StabilizaHon!of!an!uncertain!plant!over!a!rateSlimited!communicaHon!channel. bit Unstable!LTI !!!!!Sgain!bounded Small(Gain(Theorem(In(NOT(Applicable!! Achievable!inputSoutput!property!(MarHns): Unbounded! Growth!Rate Suppose s.t.

Need!for!introducing!a!pracHcal! Local!Stability!Analysis!Framework.

Recall!

" Local!stability!noHon.!

such!that

Comparison"with"existing"stabilities

InputStoSoutput!pracHcal!stability!(Jiang!et.al!1994)! Small!!!!!!signal!!!!!!!stability!is…

The!feedback!system!is!said!to!be!small!!!!!!signal!!!!!!stable!if!there! exist!!!!!!!!!!!!!!!!!!!!!!!!such!that!

Small""""""""Signal"""""""Stability

Assume!the!following!condiHons!hold.

(i)!!!!!!!!!!!!strictly!causal!&!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!such!that (ii)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!such!that (iii) (iv)

Then!the!feedback!system!is!small!!!!!!!signal!!!!!!!stable.

(Discrete"time)"Small"Level"Theorem

Small!Level!Theorem

(ConHnued)!In!parHcular,

(Discrete"time)"Small"Level"Theorem

Small!Level!Theorem

Small"Gain"Theorem"vs"Small"Level"Theorem

Small!Gain!Theorem Small!Level!Theorem

Level"Bounded"Nonlinearity

Level!bounded!nonlinearity Theorem

(i) (ii)

Assume!there!exist!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!saHsfying Then,!the!feedback!system!is!small!!!!!!!signal!!!!!!stable Suitable(for(approxima@ng(quan@za@on(errors

A"New"Local"Analysis"Framework

Small!!!!!!!signal!!!!!!!stability

Local(Boundedness

Small!Level!Theorem If!both!subsystems!have!sufficiently!small(level,(then!the!feedback! system!is!small!!!!!!!signal!!!!!!!stable.

AdenuaHon!level Input!bound

Level!Bounded!Uncertainty Suitable(for(approxima@ng(quan@za@on(errors Quan@ta@ve(Local(Analysis(Framework(based(on(Local(Boundedness.

Causal!maps Nominal!Plant!(Unstable!LTI): Uncertainty !gain!bouded

Application"Example"1

bit Unstable!LTI !!!!!Sgain!bounded

Uncertain!Plant Channel &

Application"Example"1

Unstable!LTI !!!!!Sgain!bounded

Small!Level!CondiHon

bit

If the!feedback!system!is!small!!!!!!!signal! stable!!!!!!!!!!!!!!!!!!!!!!!!!. holds!for!posiHve!constants!!!!!!!!!!!!!!!!!!!!!!!!,

( Sufficient(condi@on(on(data(rate((((((((for( the(existence(of(((((((((((((((((((((((s.t.(the(small( level(condi@on(hold.(

(Necessary(and(sufficient(condi@on(( for(scalar(nominal(plant)( (

Application"Example"1

Scalar!Nominal!Plant Theorem Assume!!!!!!!!!!!!!!!!s.t.!small!level!condiHon!holds!for!some!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!then!!!!!!!saHsfies Conversely,!if!!!!!!!saHsfies For!any!!!!!!!!!!!,!there!exist!!!!!!!!!!!!!!!!!!!!!!s.t.!nominal!part!saHsfies!the! small!level!condiHon.!

Data!rate!at!the!channel Admissible!uncertainty

Nominal!Plant (Transfer!Matrix)

TradeToff(between(data(rate(and(uncertainty

Application"Example"1

New"Class"of"Nonlinearities

Rounding!input!to!the!nearest!output.

step!size :!quanHzaHon!levels

Uniform!quanHzer gain!bounded!nonlinearity level!bounded!nonlinearity S S

EventTtriggered(sensor # ConHnuously!observes!!!!!.! # Sends!informaHon!to!the!controller!

• nly!when!!!!!saHsfies!some!condiHon.

Involves(sampling(rather(than(quan@za@on.

Application"Example"2

LTI!systems

(i)!If!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!,!then (ii)!If!!!!!!!!!!!!!!!!!!!!!,!then (iii)!If!!!!!!!!!!!!!!!!!!!!!!!!,!then

Application"Example"2

Scalar!Nominal!Plant

FixedTrange(triggered(type

Linear!system

Nonlinearity

Application"Example"2

Derive!a!condiHon!on!sensor!parameters!for!local!stability

Application"Example"2

Theorem Then,!the!eventStriggered!system!is!small!!!!!!!!!signal!!!!!!!!!stable.! In!parHcular,!!

If!

Application"Example"2

Numerical!Example

Plant Controller!1

Any(posi@ve(d(and(D(are(OK.

Stability!CondiHon

• Norm!bounds

Application"Example"2

t z(t)

• 1000

t z(t)

Application"Example"2

Numerical!Example

Plant Controller!2 Stability!CondiHon Norm!bounds

Application"Example"2

t z(t)

Conclusions

Research(Interests:!Analysis!and!Synthesis!of!Nonlinear!Systems. Nonlinear!Systems Extensions Linear!Systems

Networked!Control!Systems

1. Lyapunov!approach:!relaHon!with!internal! stabiliHes.!Focusing!on!a!bounded!band?! 2. Analysis!of!stabilizable!range!for!a!locally! stabilizing!controller.!! Possible(future(work: $ Local!analysis!framework!for! networled!control!systems! $ Extension!to!conHnuousSHme! hybrid!systems!