An H∞ LPV Design for Sampling Varying Controllers : Experimentation with a T Inverted Pendulum

a p p o r t

d e r e c h e r c h e

N 0 2 4 9 -6 3 9 9 IS R N IN R IA /R R -- 6 3 8 0 -- F R + E N G

Thème NUM

An H ∞ LPV Design for Sampling Varying Controllers : Experimentation with a T Inverted

Pendulum

David Robert — Olivier Sename — Daniel Simon

N° 6380

Décembre 2007

Centre de recherche Inria de Grenoble – Rhône-Alpes

Pendulum

David Robert

∗†

,OlivierSename

∗†

, DanielSimon

‡†

ThèmeNUMSystèmesnumériques

Équipe-ProjetNeCS

Rapportdereherhe n°6380Déembre200725pages

Abstrat: "This work has beensubmitted to the IEEE for possible publi-

ation. Copyrightmaybetransferred withoutnotie, after whih this version

maynolongerbeaessible."

This report dealswith the adaptation of a real-time ontroller's sampling

period to aountfor theavailable omputing resourevariations. Thedesign

of suh ontrollers requires aparameter-dependent disrete-time model of the

plant, where the parameteris the sampling period. A polytopi approah for

LPV(LinearParameterVarying)systemsisthendevelopedtogetan

H ^∞

^sam-

plingperioddependentontroller. Aredutionofthepolytopesize ishereper-

formed whih drastiallyredues theonservatismof theapproahand makes

easiertheontrollerimplementation. SomeexperimentalresultsonaTinverted

pendulumareprovidedtoshowtheeienyoftheapproah.

Key-words: Digitalontrol,linearparametervarying systems,

H ^∞

^ontrol,

realexperiments.

∗

GIPSA-lab(ControlSystemsDpt.),UMRINPG-CNRS5216, ENSIEG-BP 46,38402

SaintMartind'HèresCedex,Frane

†

Thisworkispartiallysupportedbythe Safe_NeCSprojetfundedbythe ANRunder

grantANR-05-SSIA-0015-03

‡

INRIARhne-Alpes,Inovallée655avenuedel'Europe,Montbonnot,38334Saint-Ismier

Cedex,Frane

H ∞

appliation à un pendule inversé

Résumé: Cerapportexamineleproblèmedel'adaptationen temps-réelde

lapérioded'éhantillonnaged'unontrleur,andeluipermettredes'adapter

aux variationsde la ressourede alul disponible. La oneptiondu ontr-

leurnéessited'avoirunmodèleentemps disretparamétréduproédé,où le

paramètre variable est la période d'éhantillonnage. Une méthode basée sur

l'approhepolytopique (LPV)estutiliséepoursynthétiserunontrleur

H ^∞

^à

périodevariable. L'utilisationd'unpolytopedetailleréduitepermetderéduire

fortementleonservatismeetlaomplexitédereonstrutionduontrleur. La

méthodeestvalidéeexpérimentalementsurunpendule inversé.

Mots-lés : Commande numérique, systèmes à paramètresvariables, om-

mande

H ^∞

^,validationexpérimentale

1 Introdution

High-tehnologyappliations(ars,householdapplianes..) areusingmoreand

moreomputing andnetwork resoures,leadingto aneed ofonsumption op-

timisationfordereasingtheostorenhaningreliabilityandperformanes. A

solutionistoimprovetheexibilityof thesystembyon-lineadaptationofthe

proessor/networkutilisation,eitherbyhangingthealgorithmorbyadapting

the sampling period. This paper deals with the latter ase and presents the

synthesisofaontrollawwithvaryingsampling period.

Few reentworkshavebeendevotedto the omputingresourevariations.

In[1℄afeedbakontrollerwithasamplingperioddependentPIDontrolleris

used. In[2,3℄afeedbakshedulerbasedonaLQoptimisationofthe ontrol

tasks periods is proposed. In [4℄ aproessor load regulation is proposed and

applied for real-timeontrolof arobot arm. Thedesign ofasampling period

dependentRSTontrollerwasproposedin[5℄. Thislatterpaperdealtwiththe

ontrol of linear SISOsystems at a variable sampling rate, and its promising

resultsalledforextensionstowardsmultivariablesystems.

The presented ontribution enhanes aprevious paper ([6℄) using a linear

parameter-varying(LPV) approahof thelinearrobustontrol framework [7℄.

The LPV approah primarily dealswith variationsof theplant's parameters,

althoughithasbeenappliedalsotoaplantparameterdependentsamplingvia

aliftingtehniqueasin[8℄.

Thispaperprovidesamethodologyfordesigningasamplingperioddepen-

dentontrollerwithperformaneadaptation,whih anbeusedin theontext

ofembeddedontrolsystems. Firstweproposeaparametriseddisretizationof

theontinuoustimeplantandoftheweightingfuntions,leadingtoadisrete-

time samplingperiod dependentaugmentedplant. Inpartiular theplantdis-

retizationapproximates thematrixexponentialbyaTaylorseriesoforder

N

^.

ThereforeweobtainapolytopiLPVmodel madeof

2 ^N

^{verties,aspresented}

in [6℄. Inthis paper we exploit thedependenybetweenthevariables param-

eters, whih are the suessivepowersof the sampling period

h, h ² , ..., h ^N

^{, to}

redue the numberof ontrollers to beombinedto

N + 1

^{. The}

H ∞

^ontrol

design method for polytopi models [7℄ is thenused to get asampling period

dependentdisrete-time ontroller. The redutionof the polytopi set drasti-

ally dereasesboththeomplexityandtheonservatismofthepreviouswork

andmakesthesolutioneasiertoimplement. Thisapproahisthenvalidatedby

experimentsonreal-time ontrolofaTinvertedpendulum.

The outline of this paper is as follows. Setion 2desribesthe plant dis-

retization and the redution of theoriginal omplexityusing theparameters

dependeny. In setion 3 the losed-loop objetives are stated and expressed

asweightingfuntions in the

H ∞

^{framework. Setion 4ommentsbriey the}

augmentedplantandgivesbakgroundon

H ∞

^{/LPVontroldesign. Theexper-}

iments onthe "T" inverted pendulum are desribed in setion 5. Finally, the

paperendswithsomeonlusionsandfurtherresearhdiretions.

2 A sample dependent LPV disrete-time model

Inthissetionthewaytoobtainapolytopidisrete-timemodel,theparameter

ofwhihbeingthesamplingperiod,isdetailed.

Weonsiderastatespaerepresentationofontinuoustime plantsas:

G :

x ˙ = Ax + Bu

y = Cx + Du

⁽¹⁾

where

x ∈ R ⁿ

^,

u ∈ R ^m

^and

y ∈ R ^p

^{. The exat} disretization of this system withazeroorderholdatthesamplingperiod

h

^leadstothedisrete-timeLPV system(2)

G d :

x k+1 = A d (h) x k + B d (h) u k

y k = C d (h) x k + D d (h) u k

(2)

with

A d = e ^Ah B d = Z h

0

e ^Aτ dτ B C d = C D d = D

(3)

Thestatespaematriesareusuallyomputedusingexpression(4)and(5) ,

see[9℄.

A d B d

0 I

= exp

A B 0 0

h

(4)

C d = C D d = D

⁽⁵⁾

with

h

^rangingin

[h min ; h max ]

^{1. Howeverin(4)}

A d

^and

B d

^arenotaneon

h

^.

2.1 Preliminary approah: Taylor expansion

Ouraim isto get apolytopi model in order to satisfyone of theframeworks

of

H ∞

^{ontrolfor LPVsystems. We herepropose to} approximate the matrix exponentialbyaTaylorseriesoforder

N

^as:

A d (h) ≈ I +

N

X

i=1

A ⁱ

i! h ⁱ

⁽⁶⁾

B d (h) ≈

N

X

i =1

A ^{i −1} B

i! h ⁱ

⁽⁷⁾

HoweveritiswellknownthattheTaylorapproximationisvalidonlyforparam-

etersnear zero. As

h

^{is assumedto belong to theinterval[}

h min

^,

h max

^℄with

h min > 0

^theapproximationwillbeonsideredaroundthenominalvalue

h 0

^of

thesampling period,as:

h = h 0 + δ

^with

h min − h 0 ≤ δ ≤ h max − h 0

⁽⁸⁾

Thenweget:

A d B d

0 I

=

A h 0 B h 0

0 I

A δ B δ

0 I

(9)

1

thevariablesamplingperiodshouldbehoseninarangewheretheontrolperformane

ishighly sensitivew.r.t. tothe samplingrate, e.g. aordingtothe ruleof thumb

ω _cl h ≈

0.2 . . . 0.6

where

ω _cl

isthedesiredlosed-loopfrequeny[9℄

where

A h 0 B h 0

0 I

= exp

A B 0 0

h 0

,

A δ B δ

0 I

= exp

A B 0 0

δ

Thisleadsto

A d = A h 0 A δ

B d = B h 0 + A h 0 B δ

(10)

Remark 1 When

δ = 0

^{, then}

A δ = I

^and

B δ = 0

^{whih means that, as}

expeted,

A d = A h 0

^and

B d = B h 0

^.

As

h 0

^{is known at design time and onstant, the Taylor}approximation is thenusedonlyfor

A δ

^and

B δ

^,as:

A d (h) ≈ A h 0 (I +

N

X

i =1

A ⁱ

i! δ ⁱ ) := A d (δ)

⁽¹¹⁾

B d (h) ≈ B h 0 + A h 0 (

N

X

i=1

A ^{i −1} B

i! δ ⁱ ) := B d (δ)

⁽¹²⁾

To evaluate the approximation error due to the Taylor approximation, a

riterionbasedonthe

H ^∞

^{norm ishosenheretoexpress theworstaseerror}

between

G d e

^and

G d

^{, both} disrete-time models using respetively the exat method and the approximated one (i.e. the Taylor series approximation of

order

N

^).

J N = max

h min <h<h max

k G d e (h, z) − G d (h, z) k ∞

⁽¹³⁾

2.2 A rst polytopi model

As

h

^{belongstotheinterval[}

h min

^,

h max

^{℄,thenweandene}

H = [δ, δ ² , . . . , δ ^N ]

thevetorof parameters.

H

^{belongsto aonvexpolytope}(hyper-polygon)

H

(14)with

2 ^N

^verties,.

H =







2 ^N

X

i =1

α i (δ)ω i : α i (δ) ≥ 0,

2 ^N

X

i =1

α i (δ) = 1







(14)

{δ, δ ² , . . . , δ ^N }, δ ⁱ ∈ {δ ⁱ _min , δ _max ⁱ }

⁽¹⁵⁾

Eahvertexisdenedbyavetor

ω i = [ν i 1 , ν i 2 , . . . , ν i N ]

^where

ν i j

^antake

theextremumvalues

{δ _min ^j

^,

δ _max ^j }

^with

δ min = h min −h 0

^and

δ max = h max −h 0

^.

The matries

A d (δ)

^and

B d (δ)

^{are therefore ane in}

H

^{and given by the}

polytopiforms:

A d (H) =

2 ^N

X

i =1

α i (δ)A d i , B d (H ) =

2 ^N

X

i =1

α i (δ)B d i

where the matries at the verties, i.e.

A d i

^and

B d i

^{, are obtained by the}

alulationof

A d (δ)

^and

B d (δ)

^{ateahvertexofthepolytope}

H

^{. Thepolytopi}

oordinates

α i

^{whih represent the position of a partiular parameter vetor}

H(δ)

^{in thepolytope}

H

^{aregivensolving:}

H (δ) =

2 ^N

X

i=1

α i (δ)ω i , α i (δ) ≥ 0 ,

2 ^N

X

i=1

α i (δ) = 1

⁽¹⁶⁾

Asanillustration,gure1showsthis transformationfor

N = 2

^with

A d 1 = A d (H 1 )

^or

H 1 = [δ min , δ _min ² ]

A d 2 = A d (H 2 )

^or

H 2 = [δ max , δ ² _min ] A d 3 = A d (H 3 )

^or

H 3 = [δ min , δ _max ² ] A d 4 = A d (H 4 )

^or

H 4 = [δ max , δ _max ² ]

H = Co {H 1 , H 2 , H 3 , H 4 }

δ min δ max

δ δ ²

δ ² _min δ ² _max

H

b b b

b b

A d (H 1 ) A d (H 2 ) A d (H 4 ) A d (H 3 )

Figure1: Exampleofpolytopefor

A d (δ)

^with

δ min = 0

Thisleadstotheplantpolytopimodel(17)where

G d i

^are

G d (H )

^evaluated

attheverties

ω i

^.

G d (H ) =

2 ^N

X

i =1

α i (δ)G d i and H =

2 ^N

X

i =1

α i (δ)ω i

⁽¹⁷⁾

Asthegain-sheduledontrollerwillbeaonvexombinationof

2 ^N

"vertex"

ontrollers, the hoie of the series order

N

^{givesa trade-o between theap-}

proximationaurayand theontrolleromplexity. Indeedoneshould notie

that:

ˆ The raw approah does not take into aount the dependene between

δ, δ ² , . . . , δ ^N

^{. Indeed,asshowningure1,thesetofparameters}

{[δ, δ ² ], 0 ≤ δ ≤ δ max }

^, represented by the paraboli urve, is inluded in the large polytopi box with 4 verties. This will of ourse indue some onser-

vatismintheontroldesign.

ˆ Moreover, when a the order of the Taylor approximation inreases, we

will see (in setion4.1) that thenumberof LMIs to besolved, whih is

2 ∗2 ^N +1

^willgrowexponentiallywhihanleadtounfeasibleoptimisation problems.

ˆ Finallytheimplementation oftheontrollerisalsodiretly linkedto the

numberofvertiesofthepolytope.

Toredue the omplexity (and the onservatismof theorresponding ontrol

designaswell),aredutionofthepolytopeisproposed below.

Remark 2 Notethat exatalulations ofmatrix exponential via diagonalisa-

tion or Cayley-Hamilton theorems are more involved here as their expression

willlead tonon anerepresentationsof

A d (H )

^and

B d (H )

^.

2.3 Redution of the polytope

It is here proposed to redue the size of the polytope using the dependeny

betweenthe suessivepowersof theparameter

δ

^{. This redutiononlystands}

for

δ min = 0

^{, whih means that}

h 0 = h min

^{is the minimal sampling period,}

i.e. relatedwithaslakonstraintonomputingresoure. Forontrolpurpose

thishoieisquitelogialasthenominalbehaviourorrespondstotheminimal

sampling period in normal situations. This period would inrease only when

omputingresoureswill belimited.

Thewaytoreduethesizeofthepolytopisetanbeseenontheexamplein

gure1,wheretheparaboliparameterslousisenlosedinthetriangledened

by

{0, 0}

^,

{δ max , 0}

^and

{δ max , δ _max ² }

^{. Thereforeitisnotneessarytoonsider}

thevertex

{0, δ ² _max }

^{to buildapolytope}enompassingtheparameterslous.

Todevelopand extendthismethod toapolytopeofsize

N

^,letuswrite:

h = h min + δ, 0 ≤ δ ≤ δ max , δ max = h max − h min ,

⁽¹⁸⁾

Thentheinequalitybelowisalwayssatised:

δ δ ⁿ ≤ δ _max ⁿ⁺¹

δ _max ⁿ δ ⁿ

^i.e.

δ ^{n +1} ≤ δ max δ ⁿ

⁽¹⁹⁾

Then it is proposed to delete the verties whih do not satisfy the above

inequality. Astheverties

H i

^of

H

^{aregivenbyavetor}

(ν 1 , ν 2 , . . . , ν N )

^where

ν i = 0

^or

δ ⁱ _max

^{aording to the onsidered vertex, then the inequality to be}

satisedisgivenby:

ν n+1 ≤ δ max ν n

⁽²⁰⁾

Thisleadstothefollowingsetof admissibleverties:

(0, 0, 0, . . . , 0) (δ max , 0, 0, . . . , 0)

(δ max , δ ² _max , 0, . . . , 0)

⁽²¹⁾

.

.

.

(δ max , δ ² _max , δ ³ _max , . . . , δ _max ^N )

Remark 3 The vertex

(0, δ ² _max , 0, . . . , 0)

^{does not satisfy inequality (20) and}

an bedisarded.

Thismethodleadstoasetof

N + 1

^{vertiesinsteadof}

2 ^N

^{. Notethatthese}

vertiesarelinearlyindependentandmakeasimplex,whihisitselfbasiallya

polytope[10℄ofminimaldimensiononsideringtheparametersspaeofdimen-

sionN.

When

N = 2

^(andfor

0 < δ < δ max

^{)thesquareisdownsizedtothetriangle}

ingure2. When

N = 3

^{thepyramidin gure3istheredutionofaube.}

Figure2: PolytoperedutionforN=2

Figure3: PolytoperedutionforN=3

3 Formulation of the

H ∞

^{/LPV ontrol problem}

Inthissetionwerstpresenttheformulationofthe

H ∞

^{ontrolproblemusing}

weighting funtion depending on the sampling period. Indeed the provided

methodologywillallowforperformaneadaptationaordingtotheomputing

resouresavailability.

The

H ∞

^{frameworkisbasedonthegeneralontrolongurationofgure4,}

where

W i

^and

W o

^{are some weighting funtions} representingthe speiation ofthedesiredlosed-loopperformanes(see[11℄). Theobjetiveisheretond

aontroller

K

^{suh internal stability is ahieved and}

k zk ˜ 2 < γk wk ˜ 2

^{, where}

γ

representsthe

H ^∞

attenuationlevel.

- - -

-

-

W o

W i

K

z

y w

u

˜

˜ z w

P P ˜

Figure4: Fousedinteronnetion

3.1 Towards disrete-time weighting funtions

Classialontroldesignassumesonstantperformaneobjetivesandprodues

a ontroller with an unique sampling period. The sampling period is hosen

aordingto theontrollerbandwidth,thenoisesensibilityandtheavailability

ofomputationresoures.Whenthesamplingperiodvariestheusableontroller

bandwidthalsovariesandthelosed-loopobjetivesshouldlogiallybeadapted.

Thereforeweproposetoadaptthebandwidthoftheweightingfuntionstothe

samplingperiod.

Themethodologyisasfollows. First

W i

^and

W o

^{aresplitintotwoparts :}

ˆ aonstantpartwithonstantpolesand zeros. This allows,forinstane,

toompensateforosillations orexible modes whih are, bydenition,

independentofthesamplingperiod.

ˆ thevariablepartontainspolesandzeroswhosepulsationsareexpressed

asananefuntionofthesamplingfrequeny

f = 1/h

^{. Thisallowsforan}

adaption of thebandwidth of the weighting funtions, and henefor an

adaption of the losed-loop performane w.r.t. the available omputing

power. These poles and zeros are here onstrained to be real by the

disretizationstep.

First of all the onstant parts of the weightingfuntions are merged with

the ontinuous-time plant model. Then a disrete-time augmented system is

developedaspresentedabove.

Thevariable part

V (s)

^{ofaweightingfuntion is thedisretizedaording}

tothefollowingmethodology:

1. fatorise

V (s)

^{as aprodut of rst order systems. We here hose poles}

andzerosdependinglinearlyofthesamplingfrequeny

f = 1/h

^,as:

V (s) = β Y

i

s − b i f

s − a i f = β Y

i

V i (s)

⁽²²⁾

with

a i , b i ∈ R

2. Considerthestatespaeobservableanonialformfor

V i (s) V i (s) :

( x ˙ i = a i f x i + f (a i − b i ) u i

y i = x i + u i

(23)

3. form the series interonnetion of the statespae representation of eah

V i (s)

^{. Thisallowstoget}

V (s)

^{oftheform(24)with}appropriatedimensions ofthestatespaematries.

V (s) :

x ˙ v = A v f x v + B v f u v

y v = C v x v + D v u v

(24)

4. Get thedisrete-time state spaerepresentation of

V (s)

^{. Thanks tothe}

anedependenein

f

^in(24)thedisrete-timemodelofthevariablepart beomesindependentof

h

^sine:



 

 

A v d = e ^{A v f h} = e ^{A v}

B v d = (A v f ) ⁻¹ (A v d − I)B v f = (A v ) ⁻¹ (A v d − I)B v

C v d = C v and D v d = D v

(25)

Remark4 Theserial interonnetionof twosystemsofthe form (26)leads to

asystemofthe form (27) .

( x ˙ = Af x + Bf u

y = C x + D u

⁽²⁶⁾

A =

A 1 0 B 2 C 1 A 2

B =

B 1

B 2 D 1

x =

x 1

x 2

C = D 2 C 1 C 2

D = D 2 D 1

(27)

Asseenmatries

C

^and

D

^{onlydepend on}

C i

^or

D i

^,

i = 1, 2

^,whihensures

that they do not depend on

f

^{. Then there is no oupling between}

A i

^and

B i

^,

i = 1, 2

^{,whih keepsthelineardependeneon}

f

^{ofthe statespae equation. As}

illustration andforward, the interonnetionof 3systemsleads toastatespae

representation (28):

A =





A 1 0 0

B 2 C 1 A 2 0 B 3 D 2 C 1 B 3 C 2 A 3



 B =



 B 1

B 2 D 1

B 3 D 2 D 1



 x =



 x 1

x 2

x 3





C = D 3 D 2 C 1 D 3 C 2 C 3

D = D 3 D 2 D 1

(28)

By iteration, the serial interonnetion of more than three systems (26) still

keepsthe form (26). Therefore the interonnetion of systems

V i (p)

^{whih are}

in form (26) leads to a system in the form (24) where the dependene on

f

makeseasier the disretizationstep.

Remark5 The simpliation between

f

^and

h

^{in (25) makes easy the dis-}

retizationstep. Thisiswhytheplantandtheweightingfuntionsareseparately

disretized, andthe augmentedplant is obtainedin disrete timeafterwards by

interonnetion. Thisisalsoaonsequeneoftheuseoftheobservableanonial

form.

3.2 The disrete-time augmented plant

Let us herepresent theoverall methodologyto get thedisrete-time plantin-

teronnetion.

Let rst onsider the followingontinuous-time model where the onstant

part of the weighting funtion

W i

^and

W o

^{has been onneted to the plant}

model:

P :



 

 

˙

x(t) = Ax(t) + B w w(t) + B u u(t) z(t) = C z x(t) + D zw w(t) + D zu u(t) y(t) = C y x(t) + D yw w(t) + D yu u(t)

(29)

where

x ∈ R ⁿ

^isthestate,

w ∈ R ^{m w}

^{representstheexogenousinputs,}

u ∈ R ^{m u}

theontrolinputs,

z ∈ R ^{p z}

^{theontrolledoutputand}

y ∈ R ^{p y}

^themeasurement vetor.

Adisrete-time representationofthe abovesystemis rstobtainedthanks

to the previous methodology. For simpliity we will note, aording to the

representation(1):

A = A B = B w B u

C = C z

C y

D =

D zw D zu

D yw D yu

(30)

Using the Taylor approximation at order

N

^{leads to apolytope}

H

^{. This}

polytopehas

r

^{verties(where}

r

^equals

2 ^N

^{forthebasiaseand}

N + 1

^forthe

redued one). Eah of the

r

^{verties is desribed by a vetor}

ω i

^{of the form}

(δ 1 , δ 2 , . . . , δ r )

^where

δ i

⁼

δ ⁱ _min

^or

δ ⁱ _max

^.

TheLPVpolytopidisrete-timemodelisgivenby:

P (H) :



 

 

x k +1 = A(H )x k + B w (H )w + B u (H )u z = C z x k + D zw w + D zu u

y = C y x k + D yw w + D yu u

(31)

H = δ δ ² . . . δ ^N

H ∈ H = Co {ω 1 , . . . , ω r } H =

r

X

i=1

α i ω i A(H) =

r

X

i=1

α i A i r

X

i=1

α i = 1 α i ≥ 0

⁽³²⁾

where, aordingtotherepresentation(2)

A d = A B d = B w B u

C d = C D d = D

⁽³³⁾

Now, the variable part of the weighting funtions

W i

^and

W o

^{are expressed}

aspreviouslypresented,whih leadsto bothdisrete-timerepresentations(34)

and(35)wherethesizeofthestatevetordependontheweightingfuntion:

W I :

( x I k+1 = A I x I k + B I w ˜ w = C I x I k + D I w ˜

(34)

W O :

( x O k+1 = A O x O k + B O z

˜

z = C O x O k + D O z

⁽³⁵⁾

Theaugmentedsystem

P ^′ (H )

^{is obtainedbythe}interonnetion of

P (H)

^,

W I

^and

W O

^{. Therefore we obtain the following LPV polytopi} disrete-time systemofstatevetor

x ^′ _k = (x k x I k x O k ) ^T

^:

P ^′ (H ) :



 

 

x ^′ _k +1 = A ^′ (H )x ^′ _k + B ^′ w (H ) ˜ w + B ^′ u (H )u

˜

z = C _z ^′ x ^′ _k + D ^′ _zw w ˜ + D ^′ _zu u y = C _y ^′ x ^′ _k + D _yw ^′ w ˜ + D _yu ^′ u

(36)

with

A ^′ (H ) =





A(H ) B w (H )C I 0

0 A I 0

B O C z B O D zw C I A O



 B _w ^′ (H ) =





B w (H)D I

B I

B O D zw D I



 B ^′ _u (H ) =



 B u (H)

0 B O D zu





C _z ^′ = D O C z D O D zw C I C O

D ^′ _zw = D O D zw D I

D ^′ _zu = D O D zu C _y ^′ = C y D yw C I 0

D ^′ _yw = D yw D I

D _yu ^′ = D yu

4 Solution to the

H ∞

^{ontrol problem for LPV}

systems

Weaimtouseherethe

H ^∞

^{ontroldesignforlinear}parameter-varyingsystems asstated in [7℄. Let the disrete-time LPV plant, mappingexogenous inputs

w

^{andontrol inputs}

u

^{to ontrolled outputs}

z

^{and measuredoutputs}

y

^{, with}

x ∈ R ^{n x}

^{,begivenbythepolytopimodel:}







x k+1 = A ^′ (H )x k + B ^′ _w (H )w + B ^′ _u (H )u z = C _z ^′ x k + D _zw ^′ w + D ^′ _zu u

y = C _y ^′ x k + D ^′ _yw w + D ^′ _yu u

(37)

wherethedependeneofthestatespaematrieson

H

^{isaneandtheparam-}

etervetor

H

^{,rangesoveraxedpolytope}

H

^with

r

^verties

ω i

H = ( _r

X

i =1

α i (δ)ω i : α i (δ) ≥ 0,

r

X

i =1

α i (δ) = 1 )

(38)

where

r

^{is equalto}

N + 1

^orto

2 ^N

^{aordingtothekindofpolytope(redued}

orfull).

4.1 Problem resolvability

Themethod onsideredhererequires thefollowingassumptions:

(A1)

D ^′ _yu (H) = 0

(A2)

B _u ^′ (H ), C ^′ _y , D ^′ _zu , D ^′ _yw

^{areparameter-}independent

(A3) the pairs

(A ^′ (H), B ^′ _u (H ))

^and

(A ^′ (H ), C _y ^′ )

^are quadratially stabilisable anddetetable over

H

respetively,

Remark6 In (37)assumption (A2)isnotsatisedduetothe

B u (H )

^termin

B ^′ _u (H )

^{. To avoid this, a stritly proper lter isadded on the ontrol input, as}

explained in [12 , 13 ℄. Itis anumerial artifat (whih of ourse inreases the

number of state variables

n e > n x

^{), therefore its bandwidth should be hosen}

high enough tobenegligible regarding theplant andobjetive bandwidths.

Proposition1 Following [7 ℄ , under the previous assumptions there exists a

gain-sheduledontroller(Figure 5 )

x K k+1 = A K (H )x K k + B K (H)y k

u k = C K (H )x K k + D K (H )y k

(39)

where

x K ∈ R ^{n e}

^{,whih ensures, overall parameter}trajetories,that :

ˆ thelosed-loop systemisinternallyquadratistable;

ˆ the

L ₂

-induednorm of the operator mapping

w

^into

z

^{isbounded by}

γ

^,

i.e.

kzk 2 < γkwk 2

if andonly ifthereexist

γ

^{andtwosymmetri matries}

(R, S)

^satisfying

2r + 1

LMIs(whih are omputedo-line) :

N R 0 0 I

T

L 1

N R 0 0 I

< 0, i = 1 . . . r

⁽⁴⁰⁾

N S 0

0 I T

L 2

N S 0 0 I

< 0, i = 1 . . . r

⁽⁴¹⁾

R I

I S

≥ 0

⁽⁴²⁾

where

L 1 =





A ¯ i R A ¯ ^T _i − R A ¯ i RC 1 ^T i B ¯ 1i

C ¯ 1i R A ¯ ^T _i −γI + ¯ C 1i R C ¯ 1 ^T i D ¯ 11i

B ¯ ^T 1 i D ¯ ^T 11 i −γI





L 2 =





A ¯ ^T _i S A ¯ i − S A ¯ ^T _i S B ¯ 1 i C ¯ _1i ^T B ¯ ^T _1i S A ¯ i −γI + B ^T _1i S B ¯ 1 i D ¯ ^T _11i

C ¯ 1i D ¯ 11i −γI





where

A ¯ i

^,

B ¯ 1i

^,

C ¯ 1i

^,

D ¯ 11i

^are

A(H) ¯

^,

B _w ^′ (H)

^,

C _z ^′ (H )

^,

D ^′ _zw (H )

^{evaluated at}

the

i ^th

^{vertexoftheparameterpolytope.}

N S

^and

N R

^{denotebasesofnullspaes}

of

( ¯ B 2 ^T , D ¯ ^T 12 )

^and

( ¯ C 2 , D ¯ 21 )

respetively.

P (H )

K(H ) -

- -

u y

˜ z

˜ w

H

Figure5: Closed-loopoftheLPVsystem

4.2 Controller reonstrution

One

R

^,

S

^and

γ

^{areobtained,theontrollersare}reonstrutedateahvertex oftheparameterpolytopeasshownin[12℄. Thegain-sheduledontroller

K(H )

isthentheonvexombinationof theseontrollers

K ( H ) :

„ A K ( H ) B K ( H ) C K ( H ) D K ( H )

«

=

r

X

i =1

α i ( δ )

„ A K _i B K _i

C K _i D K _i

«

(43)

with α i ( δ ) such that H =

r

X

i =1

α i ( δ ) ω i

(44)

Notethaton-lineshedulingoftheontrollerneedstheomputationof

α i (δ)

knowing

h

^{. Forthefullpolytopeasethepolytopioordinatesaresolutionsof}

thefollowingunder-onstrainedsystem([14,15℄):

( P ^{2 N}

i =1 α i (δ)ω i = H = [δ, δ ² , ..., δ ^N ] P 2 ^N

i=1 α i (δ) = 1, α i (δ) ≥ 0

(45)

whihanbesolvedusinganalgorithmoftheLMItoolbox[16℄. Whenthepoly-

topeis reduedto a simplex(using inequality (20)) the polytopi oordinates

aregivensolvingasimplersystem:

( P N +1

i =1 α i (δ)ω i = H = [δ, δ ² , ..., δ ^N ] P N +1

i =1 α i (δ) = 1, α i (δ) ≥ 0

⁽⁴⁶⁾

forwhihexpliitsolutionsareeasily reursivelyomputed:



 

 

α 1 = _{δ max} ^{δ max} − δ ⁻ min ^δ

α n = _δ n ^{δ n max − δ n}

max − δ ⁿ _min − P n −1

1 α i , n = [2, ..., N]

α N +1 = 1 − P N 1 α i

(47)

Thisleads, forthease

N = 2

^and

δ min = 0

^{of thenextsetion tothe simple}

expliitsolutions:

α 1 = δ max − δ δ max

, α 2 = δ _max ² − δ ²

δ _max ² − α 1 , α 3 = 1 − (α 1 + α 2 )

5 Control of the T inverted pendulum

This setionis devoted to anexperimental validation of the approah using a

"T"invertedpendulumofEduationalControlProduts 2

,availableatGIPSA-

lab,in theNeCS (Network Controlled Systems)team. These experimentswill

emphasisetheeetiveness oftheproposeddesignmethod.

2

http://www.epsystems. om/ ont rols _pe ndul um.h tm

5.1 System desription

The pendulum shown in gures6 and 7 is made of tworods. A vertialone

whihrotatesaroundthepivotaxle,andanhorizontalslidingbalaneone. Two

optionalmassesallowtomodifytheplant'sdynamialbehaviour.

Theontrolatuator(DCmotor)deliversafore

u

^{tothehorizontalsliding}

rod,throughadrivegear-rak.

The

θ

^{angle,positiveinthe}trigonometrisense,ismeasuredbytherodangle sensor. Theposition

z

^{ofthehorizontalrodismeasuredbyasensorloatedat}

themotoraxle.

TheDCmotoris torqueontrolledusing aloal urrentfeedbakloop(as-

sumedtobeasimplegainduetoitsfastdynamis). Thedynamialbehaviour

ofthesensorsisalsonegleted.

Figure6: PitureoftheTpendulum

θ(t) z(t) u(t)

Figure 7: Coordinatesof theT pendu-

lum

5.2 Modelling

A mehanialmodelof thependulum ispresentedbelow, whih takesinto a-

ountthevisousfrition(butnottheCoulombfrition).

m 1 m 1 l 0

m 1 l 0 J ¯

¨ z θ ¨

+

−f v z −m 1 z θ ˙ 2m 1 z θ ˙ 0

˙ z θ ˙

+

−m 1 sin θ

−(m 1 l 0 + m 2 l c ) sin θ − m 1 z cos θ

g = u

0

(48)

wherethetimedependeneofthestatevariablesisimpliit,andtheparameter

valuesofgivenbelowintable 1.

Table1: Parameters

Name Value Desription

m 1

^{0.217kg horizontalslidingrodmass}

m 2

^{1.795kg vertialrodmass}

l 0

^{0.33 vertialrodlength}

l c

^{-0.032m vertialrodpositionoftheentreofgravity}

g

^9.81m.s

⁻²

^{gravityaeleration}

J ¯

^0.061Nm

²

^{Nominalinertia}

f v z

^0.1kg.s

−1

visousfrition

Choosing thestatevetoras

x = [z, z, θ, ˙ θ] ˙

^{,weget thefollowingnon linear}

statespaerepresentation:



 

 

 

 

 



 

 

 

 

 



˙ x 1 = x 2

˙

x 2 = −l 0 x ˙ 4 + x 1 x ² 4 + g sin x 3 − f v z

m 1

x 2 + u m 1

˙ x 3 = x 4

˙

x 4 = 1 J 0 (x 1 ) − m 1 l ² 0

(+g(m 1 x 1 cos x 3 + m 2 l c sin x 3 )

−m 1 (l 0 x 4 + 2x 2 )x 1 x 4 − l 0 u)

(49)

with

J 0 (x 1 ) = ¯ J + m 1 x ² 1

^{. The}steady-statelinearisationaround

x = [0, 0, 0, 0]

givesthelinearstatespae representation

x(t) = ˙ Ax(t) + Bu(t)

^,

y(t) = Cx(t)

with

A =











0 1 0 0

− l 0 gm 1

J ¯ − m 1 l ² ₀ − ^f _m ^vz ₁ ⁻ J ¯ ^l − ^{0 gm} m 1 ² l ^{l 2} ₀ ^c + g 0

0 0 0 1

gm 1

J ¯ − m 1 l 0 0 J ¯ ^gm − m ² 1 ^{l c} l ² ₀ 0









 , B =









 0

l ² ₀

J ¯ − m 1 l ² 0 + _m ¹

1

0

− l 0

J ¯ − m 1 l ² ₀











C =

1 0 0 0 0 0 1 0

whihgivesnumerially:

A = 0 B B

@

0 1 0 0

− 18 . 79 − 0 . 46 14 . 82 0

0 0 0 1

56 . 92 0 −15 . 18 0 1 C C A

B = 0 B B

@ 0 7 . 52

0

−8 . 82 1 C C A

(50)

The poles of the linear model are

p 1,2 = −0.122 ± 6.784

^,

p 3 = −3.592

^and

p 4 = 3.376

^.

5.3 Performane speiation

Assuh aTpendulum systemisdiult to be ontrolled,ourmain objetive

isheretogetalosed-loopstable system,to emphasisethepratialfeasibility

oftheproposedmethodologyforreal-timeontrol.

Frompreviousexperimentswiththis plantthesampling periodis assumed

tobeintheinterval

[1, 3] ms

^.

The hosen performane objetives are represented in gure 8, where the

trakingerrorandtheontrolinputareweighted(asusualinthe

H ∞

^method-

ology).

+ d u

+ G

r K y

W u

M

W e

+

− θ

˜ e

˜ u

Figure 8: Generalontrolonguration

Thisorrespondsto thesimplemixedsensitivityproblem givenin(51).

W e (I − M S y GK 1 ) W e M S y G W u S u K 1 W u T u

_∞

≤ γ

⁽⁵¹⁾

with

K =

K 1 K 2

M =

0 0 1 0 S u = (I − K 2 G) ⁻¹ S y = (I − GK 2 ) ⁻¹

T u = −K 2 G(I − K 2 G) ⁻¹

⁽⁵²⁾

Theperformaneobjetivesarerepresentedbyweightingfuntionsandmaybe

givenbytheusualtransferfuntions [11℄:

W e (p, f) = p M S + ω S (f ) p + ω S ǫ S

ω S (f ) = h min ω S max f

⁽⁵³⁾

W u (p, f) = 1 M U

(54)

where

f = 1/h

^,

ω S max

^=1,5rad/s,

M S

^=2,

ǫ S

^=0.01and

M U

^=5.

Notiethat only

W e

^{dependsonthesamplingfrequenytoaountforper-}

formaneadaptation.

5.4 Polytopi disrete-time model

We follow herethemethodology proposed in setion2. Theapproximationis

donearoundthenominal period

h o = 1ms

^,for

h ∈ [1, 3] ms

^,i.e.

δ h ∈ [0, 2] ms

(seeRemark2).

On gure 9 the riterion (13) is evaluated for dierent sampling periods

(

h ∈ [1, 3]ms

^{) and dierent orders of the Taylor expansions (}

k ∈ [1, 5]

^{). It}

showsthatthis errormaybelargeonlyiftheorder

1

^isused.

On gure10

|G d e (δ h , z) − G d (δ h , z)|

^{is plotted aordingto the frequeny,}

evaluatedfor

5

^{samplingperiods(i.e.}

δ h ∈ [0, 2]ms

^{)andfortwoasesofTaylor}

expansions (

2

^and

4

^{). This allows to onlude that the hoie of an order}

2

ofthe Taylorexpansionis alreadyquite good asit leadsto an approximation

errorless than

−40dB

^{in the seleted sampling frequenyinterval. Note that}

hoosingthease"order2"leadstoareduedpolytopewith

3

^verties.

1 1.5 2 2.5

3 1

2 3 4 5 10 ⁻²⁰

10 ⁻¹⁵ 10 ⁻¹⁰ 10 ⁻⁵ 10 ⁰ 10 ⁵

k Erreur d’approximation : hinfnorm (G

approx −G d )

h (ms)

Erreur

Figure9: Approximationerror

5.5 LPV/

H ^∞

^design

Therststep isthedisretization of theweightingfuntions. Theaugmented

systemis got, using apreliminary rst-order lteringof theontrol input, to

satisfythedesignassumptions. Theaugmentedsystemisoforder

6

^.

Applying the designmethod developed in setion 4 leadsto the following re-

sults,ombiningtheTaylorexpansionorderandthepolytoperedution:

Polytope Nb verties

γ opt

TaylororderN=2 full 4 1.1304

TaylororderN=2 redued 3 1.1299

TaylororderN=4 full 16 1.1313

TaylororderN=4 redued 5 1.1303

Thistableemphasisesthatbothdesignoforders2and4arereliable. Forim-

plementationreasons(simpliityandomputationalomplexity)wehavehosen

theaseofthereduedpolytopeusingaTaylorexpansionoforder2.

Theorrespondingsensitivityfuntionsofthehosendesignareshowning-

ure11. Using

S e = e/r

^thesteady-statetrakingerrorislessthan

−46dB

^,with

avaryingbandwidthfrom

0.4

^to

1.2 rad/s

^{, i.e theratio3,speiedaording}

totheintervalofsamplingperiod, issatised.

10 ⁻² 10 ⁰ 10 ² 10 ⁴

−350

−300

−250

−200

−150

−100

−50 0

N=2

Magnitude (dB)

10 ⁻² 10 ⁰ 10 ² 10 ⁴

−350

−300

−250

−200

−150

−100

−50 0

N=4

Frequency (rad/s)

Magnitude (dB)

Figure10:

|G d e (δ h , z) − G d (δ h , z)|

^for

6= h

^{-Taylororder2and4}

Thepeak valueof

S u K 1

^{varies from}

1.2

^to

10.8dB

^{, whih isreasonablefor}

the ontrol gain. Note that in this partiular ase study we will benetfrom

the relativelyhigh sensitivity in high frequenies, asit allowssome persistent

ditheringin theontrolationand reduestheeet offrition, aswewill see

in theexperiments.

Finallythefuntion

M S y Gd u

^{is verylowsothat theeetofinput distur-}

bane

d u

^{onthetrakingerrorwill begreatly}attenuated.

Figure12showsthetime-domainresponseofthenonlinearpendulummodel

(angleand position)interonneted withthedisrete-timeLPVsamplingvari-

able ontroller (here for dierent frozenvaluesof the sampling periods). The

settling time varies from

1.1

^to

4.8

^{se, i.e. in aratio}

4.3

^{. Indeed weobserve}

herethegraefulandontrolleddegradationoftheperformaneduetotheadap-

tionof thesamplingdependentweightingfuntions. There isnoovershoot,as

expetedfrom thefrequenyresponsesof thesensitivityfuntion

S y GK 1

^.

5.6 Simulation results

Inthissetion,theappliationoftheproposedsamplingvariableontrollerwhen

thesamplingperiod varieson-linebetween

1

^and

3

^{mse. isprovided.}

Twoases are presented. First in gure13 the sampling period variation

is ontinuous and followsa sinusoidalsignal of frequeny

0.15rad/s

^{. Then in}

gure14somestephangesofthesamplingperiodaredone.

10 ⁻² 10 ⁰ 10 ²

−60

−40

−20 0 20

1/Wed Se

10 ⁻² 10 ⁰ 10 ²

−150

−100

−50 0 50

1/Wud SuK1

10 ⁻² 10 ⁰ 10 ²

−150

−100

−50 0 50

1/Wed MSyGdu

10 ⁻² 10 ⁰ 10 ²

−150

−100

−50 0

SyGK1 Se (e/r)

Frequency (rad/sec)

Singular Values (dB)

SuK1 (u/r)

Frequency (rad/sec)

Singular Values (dB)

MSyGdu (e/du)

Frequency (rad/sec)

Singular Values (dB)

SyGK1 (y/r)

Frequency (rad/sec)

Singular Values (dB)

Figure11: Sensitivityfuntions

0 1 2 3 4 5 6 7 8 9 10

−0.05 0 0.05 0.1 0.15 0.2

θ (rad)

Réponse indicielle du procédé non linéaire à temps continu

0 1 2 3 4 5 6 7 8 9 10

−0.06

−0.04

−0.02 0 0.02 0.04

z (m)

Temps (s)

Figure12: Responsetimeoftheontinuoustimenon-linearproess

These results show that, as expeted from the performane speiation,

thesettlingtimeofthelosed-loopsystemvariesaordinglywiththesampling

period. Whentheperiodislarge(i.eat

t = 10sec

^{)thependulumisslower,while}

whentheperiodissmall(i.eat

t = 30sec

^{inFig. 13)thependulumresponseis}

faster.Moreover,thankstotheLPVapproah,thevariations(sinusoidalorstep

hanges)ofthesamplingperioddonotleadtoabrupttransientofthependulum

behaviour. This is a great benet from the LPV approah whih ensuresthe

stability forarbitrarily fast variationsof theparameter in their allowed range

(this is due to the use of a single Lyapunov funtion in the design [7℄). The

sameassessmentanbedonefortheontrolinput.

TheLPVshemeallowsheretoguaranteethelosed-loopquadratistability,

tohaveabounded

L ₂

-induednormforallvariationofthesamplingperiodand tohaveapreditable losed-loopbehaviour.

0 10 20 30 40 50 60 70 80 90

−0.4

−0.2 0 0.2 0.4

Angle du pendule

θ [rad]

r θ

0 10 20 30 40 50 60 70 80 90

−2

−1 0 1 2

u []

Commande

0 10 20 30 40 50 60 70 80 90

1 1.5 2 2.5

3 x 10 ⁻³

h [s]

Période d’échantillonnage

Temps [s]

Figure13: MotionoftheTpendulum underasinusoidalsamplingperiod

0 10 20 30 40 50 60 70 80 90

−0.4

−0.2 0 0.2 0.4

Angle du pendule

θ [rad]

r θ

0 10 20 30 40 50 60 70 80 90

−2

−1 0 1 2

u []

Commande

0 10 20 30 40 50 60 70 80 90

0 1 2 3 4 x 10 ⁻³

h [s]

Période d’échantillonnage

Temps [s]

Figure14: MotionoftheTpendulumunder asquaresamplingperiod

5.7 Experiments

Thesenarii of theprevioussetion (simulationresults) are now implemented

forthereal plantofgure6. TheplantisontrolledthroughMatlab/Simulink

usingtheReal-time WorkshopandxPCTarget.

0 10 20 30 40 50 60 70 80 90

−0.5 0 0.5

θ [rad]

Pendulum angle

0 10 20 30 40 50 60 70 80 90

−4

−2 0 2 4

u []

Control input

0 10 20 30 40 50 60 70 80 90

1 1.5 2 2.5

3 x 10 ⁻³

Time [s]

h [s]

Sampling period

r θ

u

h

Figure15: ExperimentalmotionoftheTpendulumunderasinusoidalsampling

period

0 10 20 30 40 50 60 70 80 90

−0.5 0 0.5

θ [rad]

Pendulum angle

0 10 20 30 40 50 60 70 80 90

−5 0 5

u []

Control input

0 10 20 30 40 50 60 70 80 90

0 1 2 3 4 x 10 ⁻³

Time [s]

h [s]

h u r θ

Figure 16: Experimental motion ofthe T pendulum under asquare sampling

period

Theresultsare given in gures15 and16. Asin the previoussetion, the

settlingtimeismaximalwhenthesamplingperiodismaximal,andonversely.

Inthesameway,thereisnoabrupthangesintheontrolinput(evenwhenthe

samplingperiodabruptlyvariesfrom1to3ms asingure16).

Note that,as explainedbefore, thereal ontrol input is sensitiveto noise,

allowing to minimise the frition eet, and therefore to obtaina losed-loop

systemwithmuh lessosillations.

Finally we get similar results in simulation and experimental tests whih

showstheinherentrobustnesspropertyofthe

H ^∞

^design.

These resultsemphasise the great advantage and exibility of themethod

when the available omputing resouresmay vary, and when sampling period

variationsareusedtohandleomputingexibilitysuhasin [4℄.

6 Conlusion

In this paper, an LPV approah is proposed to design a disrete-time linear

ontrollerwithavaryingsamplingperiodandvarying performanes. Awayto

redue thepolytopefrom

2 ^N

^to

N + 1

^{verties (where}

N

^{is the Taylor order}

expansion)isprovided,whihdrastiallyreduesboththeonservatismandthe

omplexityof theresultingsampling dependentontrollerandmakesthesolu-

tion easierto implement. Further developmentsmay onernthe redutionof

theonservatismwhih isdue to tothe useaonstant Lyapunovfuntion ap-

proah,whihisknowntoprodueasub-optimalontroller. Anotherapproah

basedon [17℄is presentedin [13℄ but upto now didnotgiveimprovementsin

theresults.

Also the omplete methodology has been implemented for the ase of a

"T"invertedpendulum,whereexperimentalresultshavebeenprovided. These

resultsemphasisetherealeetivenessoftheLPVapproahaswellasitsinterest

intheontextofadaptationtovaryingproessorornetworkloadwhereabankof

swithingontrollerswouldneedtoomuhresoures. Inourase,usingasingle

ontroller synthesis,the stability andperformane property ofthe losed-loop

systemareguaranteedwhateverthespeedofvariationsofthesamplingperiod

are. In addition we also observed an interesting robustness of this ontroller

w.r.t. samplinginauraies,e.g. whihould beinduedbypreemptions ina

multi-taskingoperatingsystems.Asshowninpreliminarystudies([4,13℄),these

propertiesareofprimeinterestinthedesignofmoreomplexsystemsombining

several suhontrollersunder supervisionofafeedbak-sheduler: theontrol

periodsanbevariedarbitrarilyfastbyanoutershedulingloopunder aQoS

objetivewithnoriskofjeopardisingtheplantsstability. Howeverthespei

robustnessw.r.t. timingunertaintiesdeserveto befurtherinvestigated.

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