Periodi Sampled-Data Control for Fuzzy Systems: Intelligent Digital Redesign Approa h D.W. Kim ,Y.H. Joo ,andJ.B.Park
DepartmentofEle tri alandEle troni Engineering,YonseiUniversitySeodaemun-gu,Seoul,120-749Korea
S hoolofEle troni andInformationEngineering,KunsanNationalUniversity,Kunsan,Chonbuk,573-701Korea
Abstra t: Thispaperpresentsanewlinear-matrix-inequality-basedintelligent digitalredesign (LMI-basedIDR)te hnique
tomat hthestatesoftheanalogandthedigitalT-Sfuzzy ontrolsystemsattheintersamplinginstantsaswellasthesampling
ones. The main features of the proposed te hniqueare: 1) the aÆne ontrol s hemeis employed to in rease the degree of
freedom;2)thefuzzy-model-basedperiodi ontrolisemployed,andthe ontrolinputis hangedntimesduringonesampling
period; 3) The proposed IDR te hniqueis basedon the approximately dis retized version of the T-S fuzzysystem, but its
dis retizationerrorvanishesasnapproa hesthein nity. 4)somesuÆ ient onditions involvedinthestatemat hingand the
stabilityofthe losed-loopdis rete-timesystem anbeformulatedintheLMIsformat.
Keywords: Intelligentdigitalredesign(IDR),fuzzy-model-based ontrol,digital ontrol,fuzzysystem.
1. Introdu tion
Intelligent Digital redesign (IDR)hasgained tremendously
in reasing attention as yet another eÆ ient design tool of
sampled-datafuzzy ontrol [1℄-[6℄. TheIDRproblemis the
problem of designing a sampled-data state feedba k
on-trollersu hthatthesampled-data losed-loopfuzzysystem
isequivalenttothe ontinuous-time losed-loopfuzzysystem
inthesenseofthestatemat hing.
Therehavebeenfruitfulresear hesinthedigital ontrol
sys-temfo usingonIDRmethod.Histori ally,Jooetal. rst
at-temptedtodevelopsomeintelligentdigitalredesign
method-ologyfor omplexnonlinearsystems[1℄. Theysynergisti ally
merged both the Takagi{Sugeno (T{S) fuzzy-model-based
ontrolandthedigitalredesignte hniquefora lassof
non-linearsystems. Changetal. extendedtheintelligentdigital
redesigntoun ertainT{Sfuzzysystems[2℄. Theseapproa h
[1℄,[2℄toIDRareso alledaslo alapproa h. Thelo al
ap-proa h anallowstomat hthestatesofthe ontinuous-time
andthesampled-data losed-loopfuzzysystemsinthe
ana-lyti way,butitmaylead toundesirableand/or ina urate
results. Themajorreasonisthattheredesigneddigital
on-trolgainmatri esareobtainedby onsideringonlythelo al
state-mat hingofea hsub- losed-loopsystem[6℄. To
over- omethisweakness,Leeetal. aglobalstate-mat hing
te h-niquebasedonthe onvexoptimizationmethod,thelinear
matrixinequalities(LMIs)method,proposedin[6℄.
Spe if-i ally, their method is to globally mat h the states of the
overall losed-loopT{S fuzzysystem with the predesigned
analogfuzzy-model-based ontrollerandthosewiththe
dig-itally redesigned fuzzy-model-based ontroller, and further
toexaminethestabilizabilitybytheredesigned ontrollerin
thesenseofLyapunov. However. theIDRproblembe omes
This work was supportedinpartbythe KoreaS ien eandEngineering
Foundation(Proje tnumber:R05-2004-000-10498-0).
theoverdampedproblema ordingastransferringthelo al
approa htotheglobaloneinIDRproblem. Itmayleadto
undesirable and/orina urateresults.
AnaÆne ontrols heme[19℄ anbeanalternativebe ause
the aÆne ontrol s heme leads to in reasing the degree of
freedom. At this point, we attemptto IDR for T{S fuzzy
system based onanaÆne ontrol s hemethat has notyet
been fully ta kled underthis framework. Inaddition, the
multirate ontrol s heme [13-18℄is employedto obtain the
some advantages, whi h allows to onsider the
intersam-pling points between sampling points and to de rease the
dis retization error.
Motivatedby theabove observations,we studiesaperiodi
ontrolfor T-S fuzzysystemsbyusingtheLMI-basedIDR
method. The main features of the proposed method are
as follows: First, the aÆne ontrol s heme is employedto
in rease the degree of freedom. Se ond, the
fuzzy-model-based periodi ontrol is developed, and the ontrol input
is hanged n times during one sampling period. Se ond,
theproposedperiodi ontrols heme animprovethe
state-mat hing performan e inthe long sampling limit. Finally,
somesuÆ ient onditionsinvolvedinthestatemat hingand
the stabilityof the losed-loopdis rete-timesystem anbe
formulatedintheLMIsformat.
This paperis organized asfollows: Se tion2. ontains the
IDR problem statement of the ontinuous-time fuzzy
sys-tem. Se tion3.dis ussesthesampled-data ontroldesignfor
the ontinuous-timeT-SfuzzysystemsviatheIDRmethod.
Thispaperis on ludedinSe tion4.
ICCAS2005 June 2-5, KINTEX, Gyeonggi-Do, Korea
Consideranonlinearsystemdes ribedby _ x (t)=f(x (t);u (t)) (1) wherex(t)2R n
isthe stateve tor,and u
(t)2 R
m
is the
ontinuous-time ontrolinput,andthesubs ript\ "means
the ontinuous-time ontrol.
Tofa ilitatethe ontroldesign,wewilldevelopasimpli ed
model,whi h anrepresentthelo allinearinput{output
re-lationsof thenonlinear system. Thistypeof modelsis
re-ferred as T{S fuzzy models. The fuzzy dynami al model
orresponding to the nonlinear system (1) is des ribed by
thefollowing IF{THENrules[10℄,[11℄,[1℄,[2℄,[3℄,[6℄:
R k :IFz 1 (t)isabout k 1 and andz p (t)isabout k p , THENx_ (t)=A k x (t)+B k u (t) (2) where R k ;k 2 I q
= f1;2;:::;qg, is the kth fuzzy rule,
zr(t);r2Ip=f1;2;:::;pg,istherthpremisevariable, and
k r;(k;r) 2 IqIp, is the fuzzy set. Then, given a pair
(x
(t);u
(t)),usingthe enter-averagedefuzzi ation,
prod-u tinferen e,andsingletonfuzzi er,theoveralldynami sof
theIF-THENrules(2)hastheform
_ x (t)= q X k =1 k (z(t))(A k x (t)+B k u (t)) (3) wherek(z(t))= w k (z(t)) P q k =1 w k (z(t)) ,wk(z(t))= Q p r=1 k r(zr(t)), and k r (z r
(t)) is the grade of membership of z
r (t) in
k r .
Thepossibly time-varyingparameterve tor2R q
belongs
toa onvexpolytope,where
:= ( q X k =1 k =1; 0 k 1 )
It is lear that as varies inside , P q k =1 k(z(t))Ak and P q k =1 k (z(t))B k
rangeoveramatrixpolytope
" q X k =1 k (z(t))A k ; q X k =1 k (z(t))B k # 2Cof(A k ;B k );k2I q g
whereCo denotes the onvexhull. Inthis note,the
stabi-lizationofthepolytopi model(3)isequivalenttothe
simul-taneousstabilization ofitsverti es(Ak;Bk);k2Iq.
Inthispaper,awell- onstru ted ontinuous-timestate
feed-ba k ontroller, whi h will be employed inredesigning the
digital ontroller, is given. The ontroller is des ribed by
thefollowing IF-THENrules:
Rk:IFz1(t)isabout
k 1
and andzp(t)isabout k p,
THENu (t)= b
K
k
x (t); (4)
anditsdefuzzi edoutputis
u (t)= q X k =1 k (z(t)) b K k x (t) (5)
tal equivalent of the following ontinuous-time losed-loop
system: _ x (t)= q X k =1 q X l=1 k (z(t)) l (z(t))(A k +B k b K l )x (t) (6) 3. MainResults
3.1. Dis retization offuzzy systems
Inthefollowing,leth0 andhbethesamplingtimeandthe
ontrolupdatetime,respe tively. For onvenien e,wetake
h= h
0
N
for apositive integerN,where N isaninput
mul-tipli ity. Then, t=ih0+jhfor i2Z>0and j2Z
[0;N 1℄ ,
wheretheindexesiandjindi atesamplingand ontrol
up-dateinstants,respe tively.
Byinterfa inganidealsamplerandazero-orderholder
be-tween the plant and a ontroller, the digital fuzzy ontrol
systemisrepresentedby _ xd(t)= q X k =1 k(z(t))(Akxd(t)+Bkudk(t)): (7) where u d (t)=u d (ih 0 +jh)for t2[ih 0 +jh;ih 0 +jh+h), i2Z>0,j2Z [0;N 1℄
istheperiodi ontrolinputve tor,and
the ontrol input is hangedN times during onesampling
timeh
0
,thesubs ript\d"meansthesampled-data(digital)
ontrol.
Remark 1: Thissystem anbeviewedastheaÆne ontrol
system[19℄.
Theperiodi ontrolinputtakesthefollowing form:
udk(ih0+jh)= q X l=1 l(z(ih0+jh))Kk lxd(ih0+jh) (8) where x d (ih 0
+jh) is not required to obtain u
d (ih
0 +jh)
be auseitwill bepredi tedfromxd(ih0)afterea h ontrol
update.
Tomat hthestatesofthe ontinuous-timeandthe
sampled-data losed-loop systems, we rst have to know that the
pointwise dynami al behavior, the dis retized version of
them at every sampling and ontrol update instants.
Be- ause of the highly omplex nonlinearities among the
lin-ear subsystems, it is typi ally impossible to obtain an
ex-a t dis retized version of fuzzy system. So, the previous
approa h[6℄ is to approximate k(z(t)) as k(z(ih0+jh))
for t 2 [ih0+jh;ih0 +jh+h) so that the nonlinear
ma-tri es P q k =1 k (z(t))A k and P q k =1 k (z(t))B k an be
han-dled as the onstant matri es P
q
k =1
k(z(ih0+jh))Ak and
P q k =1 k (z(ih 0 +jh))B k .
1493
k k [ih 0 +jh;ih 0 + jh +h), i 2 Z >0 , j 2 Z [0;N 1℄ , and e P q k =1 k (z(ih 0 +jh))A k h = P q k =1 k(z(ih0+jh))e A k h ,thenthe
dis retizedsystemofthesampled-datafuzzy ontrolsystem
(7)withsamplingtimehisasfollows:
x d (ih 0 +jh+h) = q X k =1
k(z(ih0+jh))(Gkxd(ih0+jh)+Hkudk(ih0+jh))
(9) whereGk=e A k h andHk l=(Gk I)A 1 k Bl. Inordertopredi tx d
(ih0+jh)in(8) ,wewilldevelopa
gen-eralformofsolutionsto(9) ontrolledby(8)forx
d (ih
0 +jh)
withthearbitraryinitialstatexd(ih0).
Corollary1: Thesolutionto(9) losedby(8)forx
d (ih
0 +
jh)withthearbitraryinitialstatex
d (ih0)isgivenby xd(ih0+jh) = j Y v=1 ( q X k =1 q X l=1 k (z(ih0+jh vh)) l (z(ih0+jh vh)) (G k +H k K k l ))x d (ih 0 ) (10) fori2Z >0 andj2Z [1;N1℄ .
Proof: The losed-loopsystem(9)with(8)isdes ribed
by x d (ih 0 +jh+h)= q X k =1 q X l=1 k (z(ih 0 +jh)) l (z(ih 0 +jh)) (Gk+HkKk l)xd(ih0+jh) (11) Repla ingjin(11)toj 1leads xd(ih0+jh)= q X k =1 q X l=1 k(z(ih0+jh h))l(z(ih0+jh h)) (Gk+HkKk l)xd(ih0+jh h) We ompute x d (ih 0 +h)= q X k =1 q X l=1 k (z(ih 0 )) l (z(ih 0 ))(G k +H k K k l )x d (ih 0 ) x d (ih 0 +2h)= q X k =1 q X l=1 k (z(ih 0 +h)) l (z(ih 0 +h)) (G k +H k K k l )x d (ih 0 +h) = q X k 0 =1 q X l 0 =1 q X k 1 =1 q X l 1 =1 k 0 (z(ih0+h))l 0 (z(ih0+h)) k 1 (z(ih0))l 1 (z(ih0))(Gk 0 +Hk 0 Kk 0 l 0 ) (G k 1 +H k 1 K k 1 l 1 )x d (ih 0 ) for(k 0 ;j 0 ;k 1 ;j 1 ) 2I q I q | {z } 4 . Pro eeding forward, we
anreadilyobtain(10)forj>0.
d
anobtainthefollowingdis retizedversionofthe losed-loop
digitalfuzzysystemwith(7)and(8):
x d (ih 0 +jh+h) = j Y v=0 ( q X k =1 q X l=1 k(z(ih0+jh vh))l(z(ih0+jh vh)) (G k +H k K k l ))x d (ih0) (12) fori2Z>0andj2Z [0;N 1℄ .
Corollary 2: In ontinuous-time losed-loopsystem(6),
theapproximatedis rete-timemodel anbealsoobtained
as x (ih 0 +jh+h)= q X k =1 q X l=1 k (z(ih 0 +jh)) l (z(ih 0 +jh)) k lx (ih0+jh) (13) where k l =e (A k +B k b K l )h .
the solution to (13) for x
(ih
0
+jh) with the arbitrary
initialstatex (ih 0 )isgivenby x (ih 0 +jh) = j Y v=1 ( q X k =1 q X l=1 k(z(ih0+jh vh))l(z(ih0+jh vh))k l) x (ih0) (14) fori2Z>0andj2Z [1;N 1℄ .
Therefore,from(13)and(14),wedire tlyobtainthe
follow-ingdis rete-timerepresentationof (6):
x (ih0+jh+h) = j Y v=0 ( q X k =1 q X l=1 k(z(ih0+jh vh))l(z(ih0+jh vh))k l) x (ih0) (15) fori2Z>0andj2Z [0;N 1℄ .
Proof: It anbestraightforwardlyprovenbyLemma1
andCorollary1.
3.2. Design of the Periodi Control using IDR
method
The IDRproblemfor the system (7)isthe problem to
de-signaperiodi ontrollaw(8)su hthati)theorigin x=0
is aglobally asymptoti ally stable equilibriumpoint of the
losed-loopsystem _ xd(t)= q X k =1 q X l=1 k(z(t))l(z(ih0+jh)) (A k x d (t)+B k K k l x d (ih 0 +jh)); (16)
andii)by omparing(12)and(15),torealizex (ih0+jh)=
x
d (ih
0
+jh) underthe assumptionthat x
(ih 0 ) =x d (ih 0 )
, Kk l wasnumeri allysynthesized forto be aminimizerin
Theorem 2: If there exist Q = Q T
0 and onstant
matri esF
k l
su h thatthe following generalizedeigenvalue
problem(GEVP)hassolutions:
Minimize Q;F k l subje tto " Q () T k l Q G k Q H k F k l I # 0; k;l2Iq (17) " Q () T GkQ+HkFk k Q # 0; k2I q (18) " Q () T G k Q+H k F k l +G l Q+H l F lk 2 Q # 0; k;l2I q (19)
thenthestatex
d (ih
0
+jh)ofthedis rete-timerepresentation
(12) losely mat hes the dis rete-time representation (15),
and (12) is globally asymptoti ally stable in the sense of
Lyapunov, where () T
denotes the transposed element in
symmetri positions.
Proof: It anbe straightforwardly provenbyTheorem
2in[6℄
4. Con lusions
Thispaper proposed the periodi ontrol design using the
LMIapproa hfor the fuzzysystem. SomesuÆ ient
ondi-tionswerederivedforstabilizationandstatemat hingofthe
dis retizedmodelby the fastdis retization. The proposed
periodi ontrols heme animprovethestate-mat hing
per-forman einthelongsamplinglimit.
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