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Performance Analysis of Periodically Time Varying Controllers
2a-20 4
Copyrighl © 1996 IFAC I J(h Tril;!:nniul World Congress. San Francisco. USA
Performance Analysis of Periodically Time Varying Controllers* Cishen Zhangt , Jingxin Zhang 1 alld Katsuhisa Furuta# t Dt:l'a.f·t?I, en t of Elcc:tric(li (wd E l ectronic Eugtftr.cring TIt( lfnlm:, rsity of Melbourne, Pud.:ville, V;L 9052, AUSTRALIA t Eh.. df·ical Engillecr·ing Schoo l, 1lnivt rs i ly of Svuth A ,utm li(l
Whyalla CarnI"", SA 5608, AUSTRALIA # De pnrimfnt of Cont1'01 and Systems Engin e.c r;ny, Tokyo InstitTtt e of 'J'ec/moliJ9Y fJ-l ::!-! OIt.Okaya J/l. /~! Afcgll7"/J-klt, Tokyo 152, JAPAN Abstr act: This pape r lISes a fr e(llIene), .Io m a ill a pproadl 1.0 t he p e rforllHHlce analysis o f pcrio d icaUy time varying contro tlers . It. is s hown tl lat. for c.ontrol of lillea r time inva.ri a.nt plaut,s) t.h e t.ime varyillg dy nam ics o f t.he (',ont.ro ller dt'gr
1. Introductio n The research on lineal' time var yiug: COllt.WJ syste ms h as Leen active, ano JilTerclI1. res lllts It ave beel! presellte(1. For a numbe r o f control proLlcHI:>, s ll ch as tlec.ent.ralizcd control (AIl. On t.h~ ot. her ha nd , it has also been s how n thaI-linear t.irll€ vary ing c.ontro l1ers can provide n o ad va nt.ages over LTI co ntroll ers for (.O lltrol of LTI pl ant!' fe r optimal dist.1II'halJ('.e rejed.iol'J and sta.bility robust.n ess (Fe intlleh allt! Frau ds, UJ85; Khargonekar et al. , UJ85; Sharnm(\and Daltltdl , lD!H; Chapellat. and Dahleh , HH):~ ; P OO lltl and Till!; , H)87; Feuer and Goodw in , 1992 ; Zh ll ng a nd Zh:tIIg . HHJ4; alld Zha ll g ,u ltl Zhang , 1995) .
oe
It is therefOl"t: impor t.a nt. t.o gain rllrt.h ~:r uuders t.andin g of the inherent. propert.ies of t.ill lt~ varying cOIJ1.roi systems. In t.his IHI(J (~r, t.he anal.ysi8 is co nct' m ed wit.h the LPTV c.ontrol of CO Ut.iUIl01IS LT1 pl a,ut.i:i in t.erms of th e system 1/"2 alld Hoo p(:rf()rnl'llh : t~ . A fl'eqllt'llc.y do-\oV",'k SUPIWl"t etll>y A u I'> f.r .....lian Rf!Sf~; u ... h COllw:il , Ni p polI Sf-eel COI·p ., and Resca.r.-h De \·dllp nll ~ flt FUfltl 0 1" l.illiVt':l':'iit.y of S'lIIj·h A u s tntlia.
mail l approach is lH;f~ d t.o t. be analysis which is an ext.ension of t.he fre(ln ency d omain analysis of discret e periodi cally t.ime varyillg s yst.e l ns in (Feller and Goodwin , UHJ2 ; Zlt a ug and Zlmn g, 19!14; and Zhang and Zhang, 1905) . \'\' I.ile ill t he previo w; results time varying cont.roll ers include LTI controlle r;, as a subset, t he underly ing concept. (If t. I H~ allalys is in t h ,s paper is t lt aL the syst.em l.i rJIt'! va.rying performau('.e dllt' 1.0 t.he LPTV controller is distinguished from I.h ~ LTI system performance. As slIch, f.he main result. of t his paper is that t.he t.irn e varying tlyn<'llllics oft.he LPT V cO!lt,roller strictly tkgrade t.he syst.ern IJ"!. a.lld H(Y.J p e rfOTl1HnCe and an LTI con t.roller (:a ll he fo und whi ch can prov ide stric t.ly beU.er co nt.rol t.ll ar. t.he LPTV conholler. T his is in cont.rast. to t.he previoHs re:mlt.s wh~re Olle (:a.n only show) at. best ) t hat an LTl r.out.rQller is as gnod ns t.ime varying cont.rolle rs.
formulates t.h e LJ 'TV cont.rol prohlem for the analysis. Ser.t.i o l'l 3 presents t.he fr equency doma.in lift.ing all 11 prop e rt, ie~ of the lifted syst.eJn s. Th('$~ pro per t ies a re used in Sectio ll 4 to evalu ate the H't and H ov llo rJII.s of OpCII loop LPTV systems. Section 5 COIlst. rud s i lll LTI r.o ll t.rolle r for the LPTV cOlltro i sys t.em tLlLd l~v;t1 I1 t1t.es tile H J and lI o, perfoHwulCf: o f the closed lonp LPTV syst~ln . Sed.ioll
~
pt~ rfo l"lna l'l ce
2. Pl'nhlf.:1 1l1 Statelllcnt In t.his pape r , 11 · 11 d ellot.es Ihe L2 norm on continuous sig nals alld t.he L2 indu ced wmn on co nt.illllOtlS syst.em .s.
1649
and 11 · 11 00 den ote 1.1", H, and 1/ 00 norms all continuous systems, res)lp.c.t. iveiy. Let DT be t.ht! I.ime delay op erator defined as
1I·lb
w~---=-l--:z
p
y
u
wh ere y(t) is a continuolls signal a.nd T ?: O. Definitions for LTI and LPTV syst.ems are as follows.
A linear sysiem G is time
11ltJlu"i(1111
if it
satisjif!S
Figure 1: The closed loop LPTV cont.rol system JOT'
all
T ~
0
A linear sy.stem G is T.po-iodicall:lllime varyin!J if there exi~.. ts aT > 0 snch that
D, G { D,G
() )
if IIGII IS bounded. In the above d efinitions a.ll LPTV foiystem is di s tin guish ed from LTI sys tems. This is in cont,rast 1.0 the cOllventiolla l definit.ion th at. LPTV s)'!;teJIIs i tt dll d~ LT] sy.sl,ellls a.<; a. A s ystem
G is stabh'
suh~et..
It is no Led that for a T·perio dir.ally t.ime varying syi)tem G sat.isfyi ng (1), I.iu'ft! possihl y exist.s a 8f!(IIWIJ Ce o f input signa l. {ur(t)) , for I = l, 2, · ',1.0 G g ivillg lim( DTG/I./-
(;D T~ I :)
=- U
(2)
( - !'"IQ
for all 0 ~ T ~ 1'. In t. his C;\St~, l.h~ t.ime varying dynant iC'.s of the system aff~ nnt. t].('.tive but, are degenf! rated to be time invariall t with respect. t.o the itll)lJt seqllenr,e {1/./}. In regard to such all ohst!rvat.ion a nd ro r 1,lle analys is of t,be LPTV sy~t.t!ms, a ddiuit.ion for r. PTV syst em s is given
a.s A u LPTV
.!i y ~tc m
qlt C 'H:~ ujinl)llb
G IS 1!i1,tu(I,lly IIl/ IC i lt tJ(I7 ·ifJu f. to (/.
of y(t)
l,~' (/f:jinfd a~
Y(w)
=
Y.(jw). Tlt e f n 91lC ncy t'·U11.sj f-. ,"/m,,:ttMl 0[(,' i .. dcjincli flSG(W)
=
G,(jw). Th en yew) and G(w) a re
E
cOlltil1lW!1S fuuc.t.i o IIS in (-00 , +00) . Let. T :> 0 be a. t.iHl ~ pf! riOfl and WT The followillg o p eral.ors arr~ fllrt.ll t'r ddined .
Fn.:qllcn c y
,~/,if1. opr.mt07·
The H2
(w (/
H(>.) fwnns of th e LPTV sYlst em S are defi ned
Ufi
) f.T f. ~ (;( SE,8 )O( t)(SE.6,)(t)dt
T ." ."
m
S (!p
In frequen cy dornaill , SllppOSt~ t.lWl, }"3(b") is th e Laplace transform of y( t ), aud G I (,,) is th e LCI pla.ce t.ra.ns fer fll IlClion of a conti nuo us LTr sys l.elll (,'. T"tSPOIlSt:
wE(-T,+T)
TILe o r el'a t.or IVT gives all T-band limited fun ction whic.h is zero lor w :$ -7 a ud w > +7' Consi tle r t.he fet~d Lack C(!f1 t.rol syst.em ill Figure 1, where J> j!) a call sal a nd finit.e order cont.inllous LTI plant, w, z, tt, y a.rc t.he s ys t.em reCeren c:e input , cont.rolled outpuf" couho l inpu t., a.nd m ea.;:ured out.put., respectively, and f( is cl c:ansal LPTV feedback cont.ro ll er with a period T . Let S d ~ Jl o te the d osed loop sys tem wit.h t.he inJHlt. w a n(1 Lllf~ o nt.pu t z. The control prob lem unti er cons iderati o l'l can be s l,a ted as find. ng a. cont.roller b,' whi ch int.ern a ll y st.a.hilizes the syst.em an d mini m izes the eff~('. 1. of t1w ill p nt 11} on t. he out.put. z under certain perfo l'ln a ll ce Spt~ c iti cation . III t,his paper t.he analysis is cOll cerut!41 wit.h t.he syste m 11 2 an11 H oo perfor mance.
II SIl ~
T ~
{tt,(t)J ,fit .<;ut i...jit.:J (:1) flit' ft l/O :S
T.
The Jrcqu eucy
f or
otherwise
T= T T
f.
Idcltl w;utiow opeTatC} T WT in jr-r.q"t:ucy domain,
w
= "2.; .
TT.
(TTY)(W) = Y(w - '"'r ) It follows 1.10""
The time do ma.in o peratio ll associated wit.h •.II P. ope rat.or TT is the mUltiplication of y(l) hy e jwT1 . Tlllls TT is linear and T-periodically l.illW va,ryi ng.
T
=
SIp
11111 11=1
fir
IISwll
where bT = f,(t - T) is a unit.y impulse fuud,ion at. t = T, m is t,h~ input. dime nsiolJ of S, a lld E'i is Lhe illt colllmn of t.ht'! m x m ideul,il.y m atri x. In onlt
Tlu; LPTV /u; dback omtmi for' t/t G H"J, 1/, 0 1. df.fJt~n.tT!ll. c d to LTl r.ontm[ in tlte. .. r;u sc t/I(Ji S is IUd LTf, IH:r!or·)I/.{l/' (:f
I:
's
A ,~s1/.IIIJll i(ln 2: Th(: LPTV j tcdoack cont,'ol Jor th t Hoc, pn!onwm (:(; is 1/,ol (h!lf.1lfT',d{:d tv LT! contr'oi in tlte ,~f:J/.!U: t/wl S· ,:,~ uot 1Ii1·l.uu ll y· tiw.f: tnt!(H'jrmt to any se· r/1,cnu (,j" inJl ut ... iYlUlls {WI(t)} , which .:Jatisfic.s IIw,1I = 1 and at/am ... 1./".; ·.. YI.;lt:,,, H,y') )pr1a giving li mf-.o00 IISwdl
=
IISII~·
1650
It will be shown in this p II S( J(TT )112 if ,,,, IIS(Krr)lloo if and only if ASSllmpt,iol1 2 is
and it gives
sat.i~l"ficd.
va l1l es
3. Lifting ill Fl'f~queHey D OllWiu
For a p-climensional conl.jllllolls sig llal y(t) E Lz[O , 00), the norm of y(t) is
111111 .-
(00 yl'(t)y(t)dt ./0
(F - 1y)(w) =
}-. (w - kWT)
= Y(w)
.I::=- N
where Yk (w - A:Wl') is an T-band limit.ed function t.aking 011
w E [kWT -
'T , kWJ"J' + 7)'
Consider a linear syst.em G , and let U(w) and Y (w ) be t.he lifl.ed fre11ll enc.y respo nses of the input and output.• r<'sp~ct. iVt ·l y. Y(w) can be wC l t.t.en as
00 ,\'(; un
tht.~'
tlu; frequ ency domain lifled syst em i.fl defined
(1S
Cl = F(;F- 1
III frequency domain , !.he 1I01T11 of Y (w) is
It follows from Pars, ~ va l's formllla 1.lwt. t.l"lt~ norm is prt:!served under t.he frt:!quellcy 1.ransfot'lll , i.e. IIYII 111·"11·
=
The ha...,ic idea of t.he lift.ing in fn' (lllt! IlCY dOlnain is to break t.he frequ ellcy re~ poHsr;> Yew) lip int.o a. SI!qllcllce of fUll ctions and establish an eqlJivalt"llCe hdwcell Y(w) and the functiou St~qIl4 ' 1)('e , i.t>.
The lirt.ing for syst.ems es t.ablishes an equivalence bet.ween t.h~ s ystem G a mi Ul(~ lift.ed system G. Properties of t.he linial; ill fT4~~lwm(.y dOlo a in and the lifted systems a.re (kvdopcd in t.he foll owing lemmas. Due t o t.he \irnit.t~ d s pact> of tile paper , th e proofs of these lemmas are o I11it,t.~d.
L(:l1lJua 3.1: 1'h.(: non" of the signals and P1'f:SC1·1H;r! by th r; lifhng in tf,.( sense Owl
~y~te. ms
i..,
L Cl11111a 3.2: S1l1)lWSf: that G·, GUI! and Cr'fJ an: the lift ed ,'(IJ ,'ii cm s 11/ G, G,,, (HId Gf3 1 f;;sptctilJely. Then
Y(w)
w
=
2:
wl'
E (-00, +00)
wE [ - l'
F
WT
+ -)
. p~; ,.,odically
L e muUl 3.3: If G is T
~
G i.'i tint
I
time l/tl1'yi"!I, then
iUtI(U ·j l lllt .
Co ns idc.'r t.hat. G is a. linear T-periodic.ally time varyiHg sy8t.e m wit.h m -input.s a lld rI-Outputs, and its lifted sys t.t!tn is (~'. According 1.0 Lemma :3.3, G is LT! and has a freqll e llcy j.rfcr fllll Ct.i o ll G(w). A prop ert.y of G(w)
= IVTTT
where
"
Leulllla 3.4: Thf". structm'(. of th e ir·ansjn·juur.ti(nt G(w)
T'
T
tor =
(3)
T" T
1.'
,.
'1'-1
D(w) = [ Applying t.he lift.in g
op ~ rat.or
y,--,,,(-,(-,
= (FY)(w) =
[
u!(:h (,'r.:(w)
TIJ~
G:l ('" -
On C"')
+wr)
]
aU("'+"'T )
(4)
1
1.0;
(l1)
X
m transfer fUTl ction .
following lemm as fUl"t.llcr present. propert.i es of the t.ra.nsfer fllnCI.iOJlS Gc, ~: 0, ±I , ±2,"', in (4).
1, ( .... )
=
Le lllma 3.[,; TIn LT} Sy Slf ut G u is a (:a1l.'itJ.! UJ,d staif t}u; LPTV .'1y ~ tt m G is a clL1tsai'lud staM,: ."yst elll . LmullIa 3.G: A systt;tn G i ,~ [, TI if and only if th e tr·a 7L.~ In' function of the 1~f1t: (1 ,o;ys lt:rft G has (l Mock diagonlLI ,~t1'tlctun: wriilnt as bit! .'i!}slt m
Y(w)
as gt1Jen abmJt: i.5 flt:Jhl.t:d a.... til t llJlf:d In:(Juau:;y n :sponse of the sign(ti y .
The operat.or F lTIopS L:d-oo,+oo) t.o L~[-T ' +T), T-perjodi (.a ll~ · t.illle varying . The inverse of F is
and is
1'.;1
"' r) ad ... )
a :( ... - "'T) (; _1 ('"
t.o )"(w) gives tl!hf:7'(:
Y(w)
G" c... - '"'r) G_ d"') + "'r)
G_~ (,"
1~
T).
C!(w) = [
1651
<:1,( ... )
"
]
(5)
Lemma 3 .7: An EPT V syste m G with the lifted f requ ency trlln!;fer fun ction G(w) iu (4) is vil'h,ally time invariant to a sequent:t: of .. i!J71als (t/r(t)} if and mtly if
lim G.(W)ll, (W -I''''T) =
1_ 00
(6)
(I
=
for k ±l, ±2 "', alld wE (-00 , (0), vd",,,, (U,(W)) is th e frequency response of {u./(t)}. The structure of !,he lift.ed freql1 ~ lIcy t. ransfer fUIlc.t. iom; in (4) and (5) demo nst.ra.t.e that an LPT V sys t.em co nsists of mult,iple fr equency !,ra-lls fer <:h a llllelt:l wllich are t.he transfer functions Gk(W), for k =f U, on the oft' diagonal blocks of the lifted. lransf~r fllnctio n. Each of these channels shifts the input freq uency cOltlI10 ncllts by a va.lue of kWT t.o a different fr equ ellcy range rtlld t.ransfer t.he signal to the out.put. As a reslIlt t, h(~ overall transfer gain of t.he system can be aJllplifi(~ d by til e mult.ipl e fr eqll ency t.ermsfer channels . Furt.he r, Lemm a ~L 7 HI.oWl; t il e depcnd cflf_e between the LP'TV ~yst.e m time va ri a.t iolls a nd t,he llIult.ipie frequellC'.y t.ransf~r channd s. The 11II11t.ipl~ fr ctJllene,}' transfer channels art~ ill adiu(J t.o t.h ~ inp ut sig nal whenever the input. is suhject to tile hlll e varying
4. H2 and HCoJ Norms of 0IHm L oop LPTV SystnlllS In this sed.ion , th e JI'l a lul Jl<:O.) llo rlllS o f L PTV systf>ITIS a re evalua ted us ing t he eq ll ivaleuCt' het.wee n t.h~ LPTV system G ami tlH~ jirt.(~d t.ransfer flllldi oll ( ,'(w).
Theorem 4.1: T h e
f{~ 1101'111.
of
(HI.
'.PTV
.';y.~tt:tl/.
w/tCJ'f. jj rl nwto;
Proo f (out.line): The p rope rties of the syst em HOQ no rm and I.he lirt.ed t.ransfer funct.ion give
IIGlloo
=
c:
=
8111'
IIGUIl = IIGllo,
IIt'II=1
=
sup
it G(w) (8)
wE[- 'q-,~)
Since t. he maximum singll la.r value of a matrix is greater t.ha.n o r equ al 1.0 the maximulIl Binguiar valu e of any of its hl oc.k eolmnll,
~UJl uG( w) wEI-g.,g.)
This t.oget.her wit.h (8) est.ahlishes (7)
Theon~lns 4.1 ami 4.2 show that the H2 and Hoo norms of G arc ill gClJcral gre.at.e r t. ha n that of Go, and t he a mplifical.in Jl of till! Horms is duE" 1.0 the system t.ime varyin g tly uam i('.s represellj,ed by G.t.(w) o n the off diagonal blocks of t.he lined t.rallsfer fu ndion .
5. 1l ~ and JJ 'XJ PCl'fol'1uance of LPTV Co ntrol Syst.• ~IlIS
C is
I'roof(outline): Fo! i 1 ... "', let. (GE; tJ. ,)(w) I,e Hie frequency res pan.e o f (GE;6 ,)(I), U(')(w) a nd tJ. (w) be the lifted transfer fUllctiOllS of E\ awl fJ r I I·t~s p e d.ively , and Lir ::::: eWT .ir(w). It. follows from PHI'S~Va.l's forrnula a nd t.he definit.io ll of' 112 110rln, t.lL<".t
the. In(!ximtl1ll singular value of u m a-
tr'ir,
TILi s ~d.ion applies t.he p rol ·ert.ies of LPT V systems in Sect.ion :1 and 4 1.0 HIP. iLll a ly.'iis of t he H~ a.nd Hoo perfOflll a lll:t! of t.h!~ dosed loop sys tem S in Figllre L In t.he analys is, an LT] cO Il t.roll·!f J(T/ associated with t.he LPTV co nt.ro ller I< is f.()w;l.rllct.cd by usin g controller par a.met.~ riza!.iou (Youl ... f!f. a i, l!J76; Kllcera , 1974-197ti). Thf~ n the performance of t ht:' closed loop LP TV system Hsin g f{ is evaluat.ed in (:ompa.rison wit.h t.h e LTT syst.em IIsing KT ! .
IIGlll =
= '~T JoT J~:; :~;~ ,( GEii>. IO(~' )(GE,t>. )(w) d",
T IL e LTl pLw t. P in F i gul't 1 can be wri tten as
,;r IoT .r:~::, I::':,[{,' ·' (w)A . (w)],[(;(i) (w)A.(w) ].lw.L T
= "'i""ifT ' Jo rT
= "!"ttace 211"
;; .
L.l r
(9 )
IWTI' ,", '" (.,., o(.: . J )(-",)( )J A I -wT12 L..i ::::: l · ' , W (W u ,..( T
whf"!l'e P 11 , P n , P'Jl , <\11(1 P'J2 ;Lrc f,a.usai ...nd LTI syst ems. Suppose t.hat. p.J2 is sl.ahi li za ble. There exists a double (';opl'illlf:! fa dorizaf.ioll for P'22 sHch t.hat
TO ~'m {;U)·(w)c';t'J(w )dl.4lj [J«J-:'>1'/2 L.1=1
= _,'.... {WTI'I' tTU "e[C O(w) G' (w)]dw -wJT :1
= :f;. r:: trace [L: :~ _ ..., GZ(w)(.',,(w)J .Iw
P"
Theorern 4.2: Tlu: Ilr>.,J 1wnn of 111.1. LPTV 8.1/ ..;;lofl. G satisfies
(7)
= N.lr = M-I N
X [- N where N, Ai, X, Y, f,r, if, X hnd Y are causal LTI s table sys t.ellls wit.IL appropria t e dimensions. Using these , the se::t. of Iin!';u c3.<::l1al cont.roll ers which ilLt.ernally sta.bilize I
t.lw
dO~'Wtl
loop system S call be paramet.erized as
/,- = (V - MQ ,(X - NQ)- l
1652
(10)
where Q is the parameter of t.he (',ont-rolier represent.ing the set of linear (".ans a.) st.ahle syst.clIls a.nd call he wriu'<."n as (11 )
Fil];,lly, it. follows froll1 (11), (15), and ( IG) t,ha t the st.a.hle clo:ied loo p LTl syst.em g iven by the LTI cOIIt.ro ller /(7 ' / is
For a. given stabiliziug (:ollt.rollt'f f( alld t.he corresponding Q, t.he clo~:ed loop syst.em S r.nn Iw writ.t.~n in terms of Q as
whose: fre1lllency t.ransfer funct.ion is S'u(w) on the diagoTlal blocks of ,5(w).
"
( 12)
=
=
Q/.( ... - '"'T ) Q _ a( ... ) Q_:lI ... '"'T)
+
QI('" - ""Tl Q.,( ... ) Q-1 ( ... + '"'1 ' )
Q~I""
-
QuI ....
+ ""T)
S'(",) = S(I)(w) - .~(')(w)Q(w).';(''1(w) ~;!lC'" - "" 7' ) $ _IC",) S_2( ...
+ c.. r)
SI('" - ~T) .5",,( ... 1 " _ 1(-" T)
+ ....
$:,c .. -
wr)
~ 1 (w)
,:'01'"'
+ '"'1· )
1(14) ( 15)
Suppose tha.t. Qo a.nd S'o are tilt-'
iiyst( ~ m~
whose
fl't'-
qnency transfer fundio(ls
t.rcmsft'l' rUH('.I,ions Q(w) awl An LTI cOIILrolll~ r ](Tl j,,; h~ const.ructed in the fo ll o willg t.iICOl·(,lll. diagonal hlock:o; of
I)') '
papt~ r.
Theol·clll 5.2: Tilt: 1J1, 1/.o n n of tilt: LPTV system S .~(lti,<;fi(..';
(17)
if IHld only if Assumption J
1')
sutisfied.
Pl'oof: By TIJ(~orem 4 .1, t.he H 1, norm of the LPTV dos.ed loop sys t.( ~ m S il'>
1(13)
"'T)
Od ... )
Let ,S'(i)(W), for i = 1,2,3, he. tile lift.ed frequency l.rausfer function of t.he LTI systeltl S(i) ill (12). A(:~ording to Icmma3.6 . each ';·(jl(w) is a hlock di ago ll
=[
"'/ 2)Q ,..(3)
.) '
and preSt'ut. 1-he main res ults of I,he
For a giv(!n LPTV (',ont.rollf'r K. (HI) implies that. the parameter Q is LPTV. Sllppose t.hat Q is a. T-perio(lica.lly time varying syst.em. Th e lifted fn.~(l lI e u('.y l.rcUisfer flln r.t,ion of Q can he w ri t.t.e n as
=[
,..{ I) _
.) '
It is IIOW ready to provid e quantification of the H2 .uul Hoo performance of t.he closed loop LPTV syst.em
=
P II + P12 MY P~I ; S(~) P12 M ; .si3) where S(I) MP21 t and S(i) , i =: 1, 2 , :1 , are causal LTY ~t.ahle systems. The controlle·r 1( inLj~rnally st.ahilizes t.he dosed loop system if anti on ly if Q is st.able.
Q(w)
_
...")0 -
T il t!
H'J
1I0ITIl
118,oIb =
of t.he LT! system So is t1~ .I'~ _ ~ tru«(S,;(w)S.. (w)],/w
TIIlIS (17) holJs jf and o n I)' if there exist.s a k :f:. 0 sucll t.hat 8k (w) I: 0, where SJ.:(Lo.I) is a cont.innous function 1H 1.4J . O,}" LP.Hlfll
Theol'em 5.3: Till: Hf'>.J
1WNIa
{If the LPTV system S
,"nti"jit:.'" IISII~ > IIS"II= if (lUf! oulH if A.'i.'i1Uul'tiort 2 ~, $atisficd
(18)
S(w), resplx.t.ivel y.
Pr()of: It follow, from Tlteo""" 4.2 that
Theorelll 5,1: S1t l 11w SC lIwl J( i ."i (t Cafl.':;rJ/ LPTl/ CliItlrolle7· whicfl jnto1wlly ,.. t a f,ilizcs tit(; dosed itmp ,"iy.o;le1ft
Fi rstly, '"ppOse tltat. 11811", = IIS"II=. T lte" tltere exist.", "" lltence of in),lIt, signals {WI(t)) wit.1t IIwl lI = 1 such t.hat
S ;" (Ill). Then
1181100 2: IISoll=.
(j() ) is a causal LT! coutnJlh:r which, when (l]lpliu/ t.o thf 1)//I)/.t P, intenwlly .'it(lbili u;~ t/i.(; c1os~;d 10(11) .!O!I,"it.(:m SI)'
Proof: (10) implies 1.I",t. Q is LPTV if J( is LPTV , and (ll) implies t.h at. Q is a Gl.llfml sy:-:l.elU if f( is callsal. Furt,lwI , Q is a s t.able sysh>m sin eI' /\. illt.ernally s t.ahiliz.es the dosed loop syst.em . Thus , by LI' (tlllla 3.il, QII is a. causal and stabk LT I sysh~IIL wllose freqllen('.y t.ransft·r fHlldion is Qu(w) on th t~ diagoual blocks of (J(w). It, follows that. }(7' 1 givel] ill (J(i) wit.l. t.h e p:•. ralIlder Qu is a ca.usal LTI cout,roller. \Vlwll a.pplied 1.0 lhe plant. P , it gives rise t,o awl iut.erually st.
Le.t. 8TV dellot.~ t.he sY8t.em whose frequen cy transfer fllll cf,ioll is
Also , Id. ,~'II(W) be t.he lirt.ed fr, ~qllellcy transfer function of SII, aJ ld ~ j:.1.'/(w)} and {W/(;.,.o)} be t.he fre.qu ency response alld l.ltt~ liftcd freq1iell r.y r e~l"'0nse of {w/}, respectively. It. fo llows from t.he result. of Theorem 4.2 t.hal.
1653
This, together with the cOlHlit.ion lim
{·..... oo
IISIICXJ ;:;: 115'ollexn givet;
liST!' "'''I = ()
And~rsOll, B.D.O. and .1.B. Moore, (1981), Time varying feedback laws for decent,raliz~~d cont.rol, IEEE Tr-aTts. on A ut O"/l/. (d 1(: Cmi.1ml, Vo1. AC-2U, pp. 1133~1138.
In frequency domain
lirn STV(W)W,(w)
l ...... oo
=, I ...... lilll
=0
Chapellat" H. and M. DahJeh, (1992), Analysis of t.ime varying I~ollt.rol st.rategies for optimal dist.urbance reject.ion and robustness, IEEE TIans. on Automatic Control, Vol. AC-;17, pp. 1734-1745.
= ±1,±2,·
Feint.nch, A. and B.A. Franci:-;, (1985), uniformly optimal cont.rol of linear feedback sYf..t.ems, Automatica, Vo1. 21, pp. 5G3-[.74.
] W,(w)
rx,
It follows that
'_lim 00 S,(w)W,(w -
kWT) = 0,
for k
lim (S':w) - S.,(w»W,(w) = 0
1...... 00
,l
These indicat.e t.hat Ilw/ll is S\~(ll1ell('.e which attains the HCXl norm of 8, and that S is vil'lucllly time invaria.lIt t.o Ilwdl, by Lemma. 3.7. TlwrdoH~) ultder the COlHlition 1151100 ::::: IISollr:x:o the syst.elll S' doe:-: lIot. sat.isfy Assl1l11p~ tion 2.
On the other hand, suppose t.hat. IIS'II= > IIS'ulloco and that {w/(t)}, with IlwI11 =::: 1, is a seqlleuce of inplIt signals which attaills the H= nonll of S'. The inplIt se([1tence {",,J leads to
IISlloo
= 1lim __ 00
IIS'lVdl > 118.,11= 2: 1_("0...0 lirn liS., IV, II
(1$1)
It. follows that,
lim ( /j(w) - .~.,(w) IW,(w) '" 0
1...... 0<:;
Again, by Lemma 3.7) S' is not virtually time illvariant. to {WI} and, t.herefore, satisfies Assumpt.ion 2. lIence satisfied.
IISII > 11.'1,,11
R.efer(~llCCS
if ... lld ollly if ASSlIInptioll 2 is
Theorem 5.1 and Theorem 5.2 present t.he main result.s of the paper that. LTI conl.rollPfH (:;1.11 offm' stridly hd.ter control than that of LPTV cotlfrollers provided the LPTV control is not. degenerat.ed 1.0 Uw LTI perfonnaHce. G. COlleil1siol.l
In this paper a frequency dotll<'llll
Goodwin G.C. and A. Feller, (1992), Linear periodic con~ trol: A freqllenc:y domain vi,~wpoint., System.'; and Control Lftto·... , Vol. 19, pp. 379·390. In, 11, and C. Zlwug, (19tH), A mHItiratl~ digital con~ t.rollpr for model matchillg, Llutomatica Vo1. 30, No. 6, pp. 104;1-!OfiO. Kabamba, P.T., (1987), Control of linear syst.ems using generalized samplt)d dat.a. fund-ions, IEEE Tnws. (In Allt01!l.atic C:onl1"01,
Vol. AC-:32. pp.
772~78.j..
Khal'gorwkar, P.P., K. Poe,Ha, and A Tannenbaum, (1985), Robust. cont.rol of lintar time invariant. plants usiug periodic compensat.ion, lEEE Trtms. OTt Automatic (.'oll.11'o/, Vol. AC-30, pp. 1088-1096. Kucn-a, V., (UJ74-197(j), Algehraic t.heory of discret.e cont.rol for IllHlt.ivariable syslerns, Kybe.r·lIet'·ClI. Vol. 10l~, pp. 1~~40, Pllhlished ill illsta.lhnent.s. Poolla, K. and T. Ting, (19f'7), Nonlinear t,ime varying cont.rolknj for robust st.
Shamrna, .I.S. and M.A. Dahleh, (HmI), Time varying versus t,ime invariant compemmtion for reject.ioIl of persist.ent. hOllwkd disturbances and robust stabilization, IEEE TI·au.o;. on A utoH/at·ie Control, Vo1. AC~a6, pp. 8;18-847. Youla, DC., Il.A. ,labr. and .J .•1. liongiorno Jr., (1!17fi) Modmll lVieuer-Ilopf design i)f optimal controllers: part, 11, IEEE Tmn .... fin Auto1lJ.fl.t-(: Contr·o/) Vo1. AC~21, pp. ;Jl !J-:l:1S.
Zhang, C., (19!J2), A dllal rate compensat.or for system loop zero assignment., .'~hr;fnTl.5 and Conho/ Letter's, Vol. 19, pp. 2'li)~2;32. Zhang and V. Bloudel, (1993). SimuIt.aneous st.abilization Ilsiug an LTI c.ornpensat.or ;:uld a sampler and hold, Int.. .1. Control, Vol. &7, No. 2, PI', 29;3 ~ 308. Zhang, .1. and C. Zhang, (H)I)Fi), R.obust.ness analysis of (".(mt.rel syst.em using generalized sample hold functions, P7"(!(:. :1.'11(/ IEEE CDC, Orlaudo, USA. Zhang, .1. alld C. Zhang, (1 m;5), Stflhilit.y margin analysis of disnet.e periodically t.in;e varying controllers, PmL ,1411, IEEE CDC', New Orle;ws, USA.
1654
Copyrighl © 1996 IFAC I J(h Tril;!:nniul World Congress. San Francisco. USA
Performance Analysis of Periodically Time Varying Controllers* Cishen Zhangt , Jingxin Zhang 1 alld Katsuhisa Furuta# t Dt:l'a.f·t?I, en t of Elcc:tric(li (wd E l ectronic Eugtftr.cring TIt( lfnlm:, rsity of Melbourne, Pud.:ville, V;L 9052, AUSTRALIA t Eh.. df·ical Engillecr·ing Schoo l, 1lnivt rs i ly of Svuth A ,utm li(l
Whyalla CarnI"", SA 5608, AUSTRALIA # De pnrimfnt of Cont1'01 and Systems Engin e.c r;ny, Tokyo InstitTtt e of 'J'ec/moliJ9Y fJ-l ::!-! OIt.Okaya J/l. /~! Afcgll7"/J-klt, Tokyo 152, JAPAN Abstr act: This pape r lISes a fr e(llIene), .Io m a ill a pproadl 1.0 t he p e rforllHHlce analysis o f pcrio d icaUy time varying contro tlers . It. is s hown tl lat. for c.ontrol of lillea r time inva.ri a.nt plaut,s) t.h e t.ime varyillg dy nam ics o f t.he (',ont.ro ller dt'gr
1. Introductio n The research on lineal' time var yiug: COllt.WJ syste ms h as Leen active, ano JilTerclI1. res lllts It ave beel! presellte(1. For a numbe r o f control proLlcHI:>, s ll ch as tlec.ent.ralizcd control (AIl
oe
It is therefOl"t: impor t.a nt. t.o gain rllrt.h ~:r uuders t.andin g of the inherent. propert.ies of t.ill lt~ varying cOIJ1.roi systems. In t.his IHI(J (~r, t.he anal.ysi8 is co nct' m ed wit.h the LPTV c.ontrol of CO Ut.iUIl01IS LT1 pl a,ut.i:i in t.erms of th e system 1/"2 alld Hoo p(:rf()rnl'llh : t~ . A fl'eqllt'llc.y do-\oV",'k SUPIWl"t etll>y A u I'> f.r .....lian Rf!Sf~; u ... h COllw:il , Ni p polI Sf-eel COI·p ., and Resca.r.-h De \·dllp nll ~ flt FUfltl 0 1" l.illiVt':l':'iit.y of S'lIIj·h A u s tntlia.
mail l approach is lH;f~ d t.o t. be analysis which is an ext.ension of t.he fre(ln ency d omain analysis of discret e periodi cally t.ime varyillg s yst.e l ns in (Feller and Goodwin , UHJ2 ; Zlt a ug and Zlmn g, 19!14; and Zhang and Zhang, 1905) . \'\' I.ile ill t he previo w; results time varying cont.roll ers include LTI controlle r;, as a subset, t he underly ing concept. (If t. I H~ allalys is in t h ,s paper is t lt aL the syst.em l.i rJIt'! va.rying performau('.e dllt' 1.0 t.he LPTV controller is distinguished from I.h ~ LTI system performance. As slIch, f.he main result. of t his paper is that t.he t.irn e varying tlyn<'llllics oft.he LPT V cO!lt,roller strictly tkgrade t.he syst.ern IJ"!. a.lld H(Y.J p e rfOTl1HnCe and an LTI con t.roller (:a ll he fo und whi ch can prov ide stric t.ly beU.er co nt.rol t.ll ar. t.he LPTV conholler. T his is in cont.rast. to t.he previoHs re:mlt.s wh~re Olle (:a.n only show) at. best ) t hat an LTl r.out.rQller is as gnod ns t.ime varying cont.rolle rs.
formulates t.h e LJ 'TV cont.rol prohlem for the analysis. Ser.t.i o l'l 3 presents t.he fr equency doma.in lift.ing all 11 prop e rt, ie~ of the lifted syst.eJn s. Th('$~ pro per t ies a re used in Sectio ll 4 to evalu ate the H't and H ov llo rJII.s of OpCII loop LPTV systems. Section 5 COIlst. rud s i lll LTI r.o ll t.rolle r for the LPTV cOlltro i sys t.em tLlLd l~v;t1 I1 t1t.es tile H J and lI o, perfoHwulCf: o f the closed lonp LPTV syst~ln . Sed.ioll
~
pt~ rfo l"lna l'l ce
2. Pl'nhlf.:1 1l1 Statelllcnt In t.his pape r , 11 · 11 d ellot.es Ihe L2 norm on continuous sig nals alld t.he L2 indu ced wmn on co nt.illllOtlS syst.em .s.
1649
and 11 · 11 00 den ote 1.1", H, and 1/ 00 norms all continuous systems, res)lp.c.t. iveiy. Let DT be t.ht! I.ime delay op erator defined as
1I·lb
w~---=-l--:z
p
y
u
wh ere y(t) is a continuolls signal a.nd T ?: O. Definitions for LTI and LPTV syst.ems are as follows.
A linear sysiem G is time
11ltJlu"i(1111
if it
satisjif!S
Figure 1: The closed loop LPTV cont.rol system JOT'
all
T ~
0
A linear sy.stem G is T.po-iodicall:lllime varyin!J if there exi~.. ts aT > 0 snch that
D, G { D,G
() )
if IIGII IS bounded. In the above d efinitions a.ll LPTV foiystem is di s tin guish ed from LTI sys tems. This is in cont,rast 1.0 the cOllventiolla l definit.ion th at. LPTV s)'!;teJIIs i tt dll d~ LT] sy.sl,ellls a.<; a. A s ystem
G is stabh'
suh~et..
It is no Led that for a T·perio dir.ally t.ime varying syi)tem G sat.isfyi ng (1), I.iu'ft! possihl y exist.s a 8f!(IIWIJ Ce o f input signa l. {ur(t)) , for I = l, 2, · ',1.0 G g ivillg lim( DTG/I./-
(;D T~ I :)
=- U
(2)
( - !'"IQ
for all 0 ~ T ~ 1'. In t. his C;\St~, l.h~ t.ime varying dynant iC'.s of the system aff~ nnt. t].('.tive but, are degenf! rated to be time invariall t with respect. t.o the itll)lJt seqllenr,e {1/./}. In regard to such all ohst!rvat.ion a nd ro r 1,lle analys is of t,be LPTV sy~t.t!ms, a ddiuit.ion for r. PTV syst em s is given
a.s A u LPTV
.!i y ~tc m
qlt C 'H:~ ujinl)llb
G IS 1!i1,tu(I,lly IIl/ IC i lt tJ(I7 ·ifJu f. to (/.
of y(t)
l,~' (/f:jinfd a~
Y(w)
=
Y.(jw). Tlt e f n 91lC ncy t'·U11.sj f-. ,"/m,,:ttMl 0[(,' i .. dcjincli flSG(W)
=
G,(jw). Th en yew) and G(w) a re
E
cOlltil1lW!1S fuuc.t.i o IIS in (-00 , +00) . Let. T :> 0 be a. t.iHl ~ pf! riOfl and WT The followillg o p eral.ors arr~ fllrt.ll t'r ddined .
Fn.:qllcn c y
,~/,if1. opr.mt07·
The H2
(w (/
H(>.) fwnns of th e LPTV sYlst em S are defi ned
Ufi
) f.T f. ~ (;( SE,8 )O( t)(SE.6,)(t)dt
T ." ."
m
S (!p
In frequen cy dornaill , SllppOSt~ t.lWl, }"3(b") is th e Laplace transform of y( t ), aud G I (,,) is th e LCI pla.ce t.ra.ns fer fll IlClion of a conti nuo us LTr sys l.elll (,'. T"tSPOIlSt:
wE(-T,+T)
TILe o r el'a t.or IVT gives all T-band limited fun ction whic.h is zero lor w :$ -7 a ud w > +7' Consi tle r t.he fet~d Lack C(!f1 t.rol syst.em ill Figure 1, where J> j!) a call sal a nd finit.e order cont.inllous LTI plant, w, z, tt, y a.rc t.he s ys t.em reCeren c:e input , cont.rolled outpuf" couho l inpu t., a.nd m ea.;:ured out.put., respectively, and f( is cl c:ansal LPTV feedback cont.ro ll er with a period T . Let S d ~ Jl o te the d osed loop sys tem wit.h t.he inJHlt. w a n(1 Lllf~ o nt.pu t z. The control prob lem unti er cons iderati o l'l can be s l,a ted as find. ng a. cont.roller b,' whi ch int.ern a ll y st.a.hilizes the syst.em an d mini m izes the eff~('. 1. of t1w ill p nt 11} on t. he out.put. z under certain perfo l'ln a ll ce Spt~ c iti cation . III t,his paper t.he analysis is cOll cerut!41 wit.h t.he syste m 11 2 an11 H oo perfor mance.
II SIl ~
T ~
{tt,(t)J ,fit .<;ut i...jit.:J (:1) flit' ft l/O :S
T.
The Jrcqu eucy
f or
otherwise
T= T T
f.
Idcltl w;utiow opeTatC} T WT in jr-r.q"t:ucy domain,
w
= "2.; .
TT.
(TTY)(W) = Y(w - '"'r ) It follows 1.10""
The time do ma.in o peratio ll associated wit.h •.II P. ope rat.or TT is the mUltiplication of y(l) hy e jwT1 . Tlllls TT is linear and T-periodically l.illW va,ryi ng.
T
=
SIp
11111 11=1
fir
IISwll
where bT = f,(t - T) is a unit.y impulse fuud,ion at. t = T, m is t,h~ input. dime nsiolJ of S, a lld E'i is Lhe illt colllmn of t.ht'! m x m ideul,il.y m atri x. In onlt
Tlu; LPTV /u; dback omtmi for' t/t G H"J, 1/, 0 1. df.fJt~n.tT!ll. c d to LTl r.ontm[ in tlte. .. r;u sc t/I(Ji S is IUd LTf, IH:r!or·)I/.{l/' (:f
I:
's
A ,~s1/.IIIJll i(ln 2: Th(: LPTV j tcdoack cont,'ol Jor th t Hoc, pn!onwm (:(; is 1/,ol (h!lf.1lfT',d{:d tv LT! contr'oi in tlte ,~f:J/.!U: t/wl S· ,:,~ uot 1Ii1·l.uu ll y· tiw.f: tnt!(H'jrmt to any se· r/1,cnu (,j" inJl ut ... iYlUlls {WI(t)} , which .:Jatisfic.s IIw,1I = 1 and at/am ... 1./".; ·.. YI.;lt:,,, H,y') )pr1a giving li mf-.o00 IISwdl
=
IISII~·
1650
It will be shown in this p
and it gives
sat.i~l"ficd.
va l1l es
3. Lifting ill Fl'f~queHey D OllWiu
For a p-climensional conl.jllllolls sig llal y(t) E Lz[O , 00), the norm of y(t) is
111111 .-
(00 yl'(t)y(t)dt ./0
(F - 1y)(w) =
}-. (w - kWT)
= Y(w)
.I::=- N
where Yk (w - A:Wl') is an T-band limit.ed function t.aking 011
w E [kWT -
'T , kWJ"J' + 7)'
Consider a linear syst.em G , and let U(w) and Y (w ) be t.he lifl.ed fre11ll enc.y respo nses of the input and output.• r<'sp~ct. iVt ·l y. Y(w) can be wC l t.t.en as
00 ,\'(; un
tht.~'
tlu; frequ ency domain lifled syst em i.fl defined
(1S
Cl = F(;F- 1
III frequency domain , !.he 1I01T11 of Y (w) is
It follows from Pars, ~ va l's formllla 1.lwt. t.l"lt~ norm is prt:!served under t.he frt:!quellcy 1.ransfot'lll , i.e. IIYII 111·"11·
=
The ha...,ic idea of t.he lift.ing in fn' (lllt! IlCY dOlnain is to break t.he frequ ellcy re~ poHsr;> Yew) lip int.o a. SI!qllcllce of fUll ctions and establish an eqlJivalt"llCe hdwcell Y(w) and the functiou St~qIl4 ' 1)('e , i.t>.
The lirt.ing for syst.ems es t.ablishes an equivalence bet.ween t.h~ s ystem G a mi Ul(~ lift.ed system G. Properties of t.he linial; ill fT4~~lwm(.y dOlo a in and the lifted systems a.re (kvdopcd in t.he foll owing lemmas. Due t o t.he \irnit.t~ d s pact> of tile paper , th e proofs of these lemmas are o I11it,t.~d.
L(:l1lJua 3.1: 1'h.(: non" of the signals and P1'f:SC1·1H;r! by th r; lifhng in tf,.( sense Owl
~y~te. ms
i..,
L Cl11111a 3.2: S1l1)lWSf: that G·, GUI! and Cr'fJ an: the lift ed ,'(IJ ,'ii cm s 11/ G, G,,, (HId Gf3 1 f;;sptctilJely. Then
Y(w)
w
=
2:
wl'
E (-00, +00)
wE [ - l'
F
WT
+ -)
. p~; ,.,odically
L e muUl 3.3: If G is T
~
G i.'i tint
I
time l/tl1'yi"!I, then
iUtI(U ·j l lllt .
Co ns idc.'r t.hat. G is a. linear T-periodic.ally time varyiHg sy8t.e m wit.h m -input.s a lld rI-Outputs, and its lifted sys t.t!tn is (~'. According 1.0 Lemma :3.3, G is LT! and has a freqll e llcy j.r
= IVTTT
where
"
Leulllla 3.4: Thf". structm'(. of th e ir·ansjn·juur.ti(nt G(w)
T'
T
tor =
(3)
T" T
1.'
,.
'1'-1
D(w) = [ Applying t.he lift.in g
op ~ rat.or
y,--,,,(-,(-,
= (FY)(w) =
[
u!(:h (,'r.:(w)
TIJ~
G:l ('" -
On C"')
+wr)
]
aU("'+"'T )
(4)
1
1.0;
(l1)
X
m transfer fUTl ction .
following lemm as fUl"t.llcr present. propert.i es of the t.ra.nsfer fllnCI.iOJlS Gc, ~: 0, ±I , ±2,"', in (4).
1, ( .... )
=
Le lllma 3.[,; TIn LT} Sy Slf ut G u is a (:a1l.'itJ.! UJ,d staif t}u; LPTV .'1y ~ tt m G is a clL1tsai'lud staM,: ."yst elll . LmullIa 3.G: A systt;tn G i ,~ [, TI if and only if th e tr·a 7L.~ In' function of the 1~f1t: (1 ,o;ys lt:rft G has (l Mock diagonlLI ,~t1'tlctun: wriilnt as bit! .'i!}slt m
Y(w)
as gt1Jen abmJt: i.5 flt:Jhl.t:d a.... til t llJlf:d In:(Juau:;y n :sponse of the sign(ti y .
The operat.or F lTIopS L:d-oo,+oo) t.o L~[-T ' +T), T-perjodi (.a ll~ · t.illle varying . The inverse of F is
and is
1'.;1
"' r) ad ... )
a :( ... - "'T) (; _1 ('"
t.o )"(w) gives tl!hf:7'(:
Y(w)
G" c... - '"'r) G_ d"') + "'r)
G_~ (,"
1~
T).
C!(w) = [
1651
<:1,( ... )
"
]
(5)
Lemma 3 .7: An EPT V syste m G with the lifted f requ ency trlln!;fer fun ction G(w) iu (4) is vil'h,ally time invariant to a sequent:t: of .. i!J71als (t/r(t)} if and mtly if
lim G.(W)ll, (W -I''''T) =
1_ 00
(6)
(I
=
for k ±l, ±2 "', alld wE (-00 , (0), vd",,,, (U,(W)) is th e frequency response of {u./(t)}. The structure of !,he lift.ed freql1 ~ lIcy t. ransfer fUIlc.t. iom; in (4) and (5) demo nst.ra.t.e that an LPT V sys t.em co nsists of mult,iple fr equency !,ra-lls fer <:h a llllelt:l wllich are t.he transfer functions Gk(W), for k =f U, on the oft' diagonal blocks of the lifted. lransf~r fllnctio n. Each of these channels shifts the input freq uency cOltlI10 ncllts by a va.lue of kWT t.o a different fr equ ellcy range rtlld t.ransfer t.he signal to the out.put. As a reslIlt t, h(~ overall transfer gain of t.he system can be aJllplifi(~ d by til e mult.ipl e fr eqll ency t.ermsfer channels . Furt.he r, Lemm a ~L 7 HI.oWl; t il e depcnd cflf_e between the LP'TV ~yst.e m time va ri a.t iolls a nd t,he llIult.ipie frequellC'.y t.ransf~r channd s. The 11II11t.ipl~ fr ctJllene,}' transfer channels art~ ill adiu(J t.o t.h ~ inp ut sig nal whenever the input. is suhject to tile hlll e varying
4. H2 and HCoJ Norms of 0IHm L oop LPTV SystnlllS In this sed.ion , th e JI'l a lul Jl<:O.) llo rlllS o f L PTV systf>ITIS a re evalua ted us ing t he eq ll ivaleuCt' het.wee n t.h~ LPTV system G ami tlH~ jirt.(~d t.ransfer flllldi oll ( ,'(w).
Theorem 4.1: T h e
f{~ 1101'111.
of
(HI.
'.PTV
.';y.~tt:tl/.
w/tCJ'f. jj rl nwto;
Proo f (out.line): The p rope rties of the syst em HOQ no rm and I.he lirt.ed t.ransfer funct.ion give
IIGlloo
=
c:
=
8111'
IIGUIl = IIGllo,
IIt'II=1
=
sup
it G(w) (8)
wE[- 'q-,~)
Since t. he maximum singll la.r value of a matrix is greater t.ha.n o r equ al 1.0 the maximulIl Binguiar valu e of any of its hl oc.k eolmnll,
~UJl uG( w) wEI-g.,g.)
This t.oget.her wit.h (8) est.ahlishes (7)
Theon~lns 4.1 ami 4.2 show that the H2 and Hoo norms of G arc ill gClJcral gre.at.e r t. ha n that of Go, and t he a mplifical.in Jl of till! Horms is duE" 1.0 the system t.ime varyin g tly uam i('.s represellj,ed by G.t.(w) o n the off diagonal blocks of t.he lined t.rallsfer fu ndion .
5. 1l ~ and JJ 'XJ PCl'fol'1uance of LPTV Co ntrol Syst.• ~IlIS
C is
I'roof(outline): Fo! i 1 ... "', let. (GE; tJ. ,)(w) I,e Hie frequency res pan.e o f (GE;6 ,)(I), U(')(w) a nd tJ. (w) be the lifted transfer fUllctiOllS of E\ awl fJ r I I·t~s p e d.ively , and Lir ::::: eWT .ir(w). It. follows from PHI'S~Va.l's forrnula a nd t.he definit.io ll of' 112 110rln, t.lL<".t
the. In(!ximtl1ll singular value of u m a-
tr'ir,
TILi s ~d.ion applies t.he p rol ·ert.ies of LPT V systems in Sect.ion :1 and 4 1.0 HIP. iLll a ly.'iis of t he H~ a.nd Hoo perfOflll a lll:t! of t.h!~ dosed loop sys tem S in Figllre L In t.he analys is, an LT] cO Il t.roll·!f J(T/ associated with t.he LPTV co nt.ro ller I< is f.()w;l.rllct.cd by usin g controller par a.met.~ riza!.iou (Youl ... f!f. a i, l!J76; Kllcera , 1974-197ti). Thf~ n the performance of t ht:' closed loop LP TV system Hsin g f{ is evaluat.ed in (:ompa.rison wit.h t.h e LTT syst.em IIsing KT ! .
IIGlll =
= '~T JoT J~:; :~;~ ,( GEii>. IO(~' )(GE,t>. )(w) d",
T IL e LTl pLw t. P in F i gul't 1 can be wri tten as
,;r IoT .r:~::, I::':,[{,' ·' (w)A . (w)],[(;(i) (w)A.(w) ].lw.L T
= "'i""ifT ' Jo rT
= "!"ttace 211"
;; .
L.l r
(9 )
IWTI' ,", '" (.,., o(.: . J )(-",)( )J A I -wT12 L..i ::::: l · ' , W (W u ,..( T
whf"!l'e P 11 , P n , P'Jl , <\11(1 P'J2 ;Lrc f,a.usai ...nd LTI syst ems. Suppose t.hat. p.J2 is sl.ahi li za ble. There exists a double (';opl'illlf:! fa dorizaf.ioll for P'22 sHch t.hat
TO ~'m {;U)·(w)c';t'J(w )dl.4lj [J«J-:'>1'/2 L.1=1
= _,'.... {WTI'I' tTU "e[C O(w) G' (w)]dw -wJT :1
= :f;. r:: trace [L: :~ _ ..., GZ(w)(.',,(w)J .Iw
P"
Theorern 4.2: Tlu: Ilr>.,J 1wnn of 111.1. LPTV 8.1/ ..;;lofl. G satisfies
(7)
= N.lr = M-I N
X [- N where N, Ai, X, Y, f,r, if, X hnd Y are causal LTI s table sys t.ellls wit.IL appropria t e dimensions. Using these , the se::t. of Iin!';u c3.<::l1al cont.roll ers which ilLt.ernally sta.bilize I
t.lw
dO~'Wtl
loop system S call be paramet.erized as
/,- = (V - MQ ,(X - NQ)- l
1652
(10)
where Q is the parameter of t.he (',ont-rolier represent.ing the set of linear (".ans a.) st.ahle syst.clIls a.nd call he wriu'<."n as (11 )
Fil];,lly, it. follows froll1 (11), (15), and ( IG) t,ha t the st.a.hle clo:ied loo p LTl syst.em g iven by the LTI cOIIt.ro ller /(7 ' / is
For a. given stabiliziug (:ollt.rollt'f f( alld t.he corresponding Q, t.he clo~:ed loop syst.em S r.nn Iw writ.t.~n in terms of Q as
whose: fre1lllency t.ransfer funct.ion is S'u(w) on the diagoTlal blocks of ,5(w).
"
( 12)
=
=
Q/.( ... - '"'T ) Q _ a( ... ) Q_:lI ... '"'T)
+
QI('" - ""Tl Q.,( ... ) Q-1 ( ... + '"'1 ' )
Q~I""
-
QuI ....
+ ""T)
S'(",) = S(I)(w) - .~(')(w)Q(w).';(''1(w) ~;!lC'" - "" 7' ) $ _IC",) S_2( ...
+ c.. r)
SI('" - ~T) .5",,( ... 1 " _ 1(-" T)
+ ....
$:,c .. -
wr)
~ 1 (w)
,:'01'"'
+ '"'1· )
1(14) ( 15)
Suppose tha.t. Qo a.nd S'o are tilt-'
iiyst( ~ m~
whose
fl't'-
qnency transfer fundio(ls
t.rcmsft'l' rUH('.I,ions Q(w) awl An LTI cOIILrolll~ r ](Tl j,,; h~ const.ructed in the fo ll o willg t.iICOl·(,lll. diagonal hlock:o; of
I)') '
papt~ r.
Theol·clll 5.2: Tilt: 1J1, 1/.o n n of tilt: LPTV system S .~(lti,<;fi(..';
(17)
if IHld only if Assumption J
1')
sutisfied.
Pl'oof: By TIJ(~orem 4 .1, t.he H 1, norm of the LPTV dos.ed loop sys t.( ~ m S il'>
1(13)
"'T)
Od ... )
Let ,S'(i)(W), for i = 1,2,3, he. tile lift.ed frequency l.rausfer function of t.he LTI systeltl S(i) ill (12). A(:~ording to Icmma3.6 . each ';·(jl(w) is a hlock di ago ll
=[
"'/ 2)Q ,..(3)
.) '
and preSt'ut. 1-he main res ults of I,he
For a giv(!n LPTV (',ont.rollf'r K. (HI) implies that. the parameter Q is LPTV. Sllppose t.hat Q is a. T-perio(lica.lly time varying syst.em. Th e lifted fn.~(l lI e u('.y l.rcUisfer flln r.t,ion of Q can he w ri t.t.e n as
=[
,..{ I) _
.) '
It is IIOW ready to provid e quantification of the H2 .uul Hoo performance of t.he closed loop LPTV syst.em
=
P II + P12 MY P~I ; S(~) P12 M ; .si3) where S(I) MP21 t and S(i) , i =: 1, 2 , :1 , are causal LTY ~t.ahle systems. The controlle·r 1( inLj~rnally st.ahilizes t.he dosed loop system if anti on ly if Q is st.able.
Q(w)
_
...")0 -
T il t!
H'J
1I0ITIl
118,oIb =
of t.he LT! system So is t1~ .I'~ _ ~ tru«(S,;(w)S.. (w)],/w
TIIlIS (17) holJs jf and o n I)' if there exist.s a k :f:. 0 sucll t.hat 8k (w) I: 0, where SJ.:(Lo.I) is a cont.innous function 1H 1.4J . O,}" LP.Hlfll
Theol'em 5.3: Till: Hf'>.J
1WNIa
{If the LPTV system S
,"nti"jit:.'" IISII~ > IIS"II= if (lUf! oulH if A.'i.'i1Uul'tiort 2 ~, $atisficd
(18)
S(w), resplx.t.ivel y.
Pr()of: It follow, from Tlteo""" 4.2 that
Theorelll 5,1: S1t l 11w SC lIwl J( i ."i (t Cafl.':;rJ/ LPTl/ CliItlrolle7· whicfl jnto1wlly ,.. t a f,ilizcs tit(; dosed itmp ,"iy.o;le1ft
Fi rstly, '"ppOse tltat. 11811", = IIS"II=. T lte" tltere exist.", "" lltence of in),lIt, signals {WI(t)) wit.1t IIwl lI = 1 such t.hat
S ;" (Ill). Then
1181100 2: IISoll=.
(j() ) is a causal LT! coutnJlh:r which, when (l]lpliu/ t.o thf 1)//I)/.t P, intenwlly .'it(lbili u;~ t/i.(; c1os~;d 10(11) .!O!I,"it.(:m SI)'
Proof: (10) implies 1.I",t. Q is LPTV if J( is LPTV , and (ll) implies t.h at. Q is a Gl.llfml sy:-:l.elU if f( is callsal. Furt,lwI , Q is a s t.able sysh>m sin eI' /\. illt.ernally s t.ahiliz.es the dosed loop syst.em . Thus , by LI' (tlllla 3.il, QII is a. causal and stabk LT I sysh~IIL wllose freqllen('.y t.ransft·r fHlldion is Qu(w) on th t~ diagoual blocks of (J(w). It, follows that. }(7' 1 givel] ill (J(i) wit.l. t.h e p:•. ralIlder Qu is a ca.usal LTI cout,roller. \Vlwll a.pplied 1.0 lhe plant. P , it gives rise t,o awl iut.erually st.
Le.t. 8TV dellot.~ t.he sY8t.em whose frequen cy transfer fllll cf,ioll is
Also , Id. ,~'II(W) be t.he lirt.ed fr, ~qllellcy transfer function of SII, aJ ld ~ j:.1.'/(w)} and {W/(;.,.o)} be t.he fre.qu ency response alld l.ltt~ liftcd freq1iell r.y r e~l"'0nse of {w/}, respectively. It. fo llows from t.he result. of Theorem 4.2 t.hal.
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This, together with the cOlHlit.ion lim
{·..... oo
IISIICXJ ;:;: 115'ollexn givet;
liST!' "'''I = ()
And~rsOll, B.D.O. and .1.B. Moore, (1981), Time varying feedback laws for decent,raliz~~d cont.rol, IEEE Tr-aTts. on A ut O"/l/. (d 1(: Cmi.1ml, Vo1. AC-2U, pp. 1133~1138.
In frequency domain
lirn STV(W)W,(w)
l ...... oo
=, I ...... lilll
=0
Chapellat" H. and M. DahJeh, (1992), Analysis of t.ime varying I~ollt.rol st.rategies for optimal dist.urbance reject.ion and robustness, IEEE TIans. on Automatic Control, Vol. AC-;17, pp. 1734-1745.
= ±1,±2,·
Feint.nch, A. and B.A. Franci:-;, (1985), uniformly optimal cont.rol of linear feedback sYf..t.ems, Automatica, Vo1. 21, pp. 5G3-[.74.
] W,(w)
rx,
It follows that
'_lim 00 S,(w)W,(w -
kWT) = 0,
for k
lim (S':w) - S.,(w»W,(w) = 0
1...... 00
,l
These indicat.e t.hat Ilw/ll is S\~(ll1ell('.e which attains the HCXl norm of 8, and that S is vil'lucllly time invaria.lIt t.o Ilwdl, by Lemma. 3.7. TlwrdoH~) ultder the COlHlition 1151100 ::::: IISollr:x:o the syst.elll S' doe:-: lIot. sat.isfy Assl1l11p~ tion 2.
On the other hand, suppose t.hat. IIS'II= > IIS'ulloco and that {w/(t)}, with IlwI11 =::: 1, is a seqlleuce of inplIt signals which attaills the H= nonll of S'. The inplIt se([1tence {",,J leads to
IISlloo
= 1lim __ 00
IIS'lVdl > 118.,11= 2: 1_("0...0 lirn liS., IV, II
(1$1)
It. follows that,
lim ( /j(w) - .~.,(w) IW,(w) '" 0
1...... 0<:;
Again, by Lemma 3.7) S' is not virtually time illvariant. to {WI} and, t.herefore, satisfies Assumpt.ion 2. lIence satisfied.
IISII > 11.'1,,11
R.efer(~llCCS
if ... lld ollly if ASSlIInptioll 2 is
Theorem 5.1 and Theorem 5.2 present t.he main result.s of the paper that. LTI conl.rollPfH (:;1.11 offm' stridly hd.ter control than that of LPTV cotlfrollers provided the LPTV control is not. degenerat.ed 1.0 Uw LTI perfonnaHce. G. COlleil1siol.l
In this paper a frequency dotll<'llll
Goodwin G.C. and A. Feller, (1992), Linear periodic con~ trol: A freqllenc:y domain vi,~wpoint., System.'; and Control Lftto·... , Vol. 19, pp. 379·390. In, 11, and C. Zlwug, (19tH), A mHItiratl~ digital con~ t.rollpr for model matchillg, Llutomatica Vo1. 30, No. 6, pp. 104;1-!OfiO. Kabamba, P.T., (1987), Control of linear syst.ems using generalized samplt)d dat.a. fund-ions, IEEE Tnws. (In Allt01!l.atic C:onl1"01,
Vol. AC-:32. pp.
772~78.j..
Khal'gorwkar, P.P., K. Poe,Ha, and A Tannenbaum, (1985), Robust. cont.rol of lintar time invariant. plants usiug periodic compensat.ion, lEEE Trtms. OTt Automatic (.'oll.11'o/, Vol. AC-30, pp. 1088-1096. Kucn-a, V., (UJ74-197(j), Algehraic t.heory of discret.e cont.rol for IllHlt.ivariable syslerns, Kybe.r·lIet'·ClI. Vol. 10l~, pp. 1~~40, Pllhlished ill illsta.lhnent.s. Poolla, K. and T. Ting, (19f'7), Nonlinear t,ime varying cont.rolknj for robust st.
Shamrna, .I.S. and M.A. Dahleh, (HmI), Time varying versus t,ime invariant compemmtion for reject.ioIl of persist.ent. hOllwkd disturbances and robust stabilization, IEEE TI·au.o;. on A utoH/at·ie Control, Vo1. AC~a6, pp. 8;18-847. Youla, DC., Il.A. ,labr. and .J .•1. liongiorno Jr., (1!17fi) Modmll lVieuer-Ilopf design i)f optimal controllers: part, 11, IEEE Tmn .... fin Auto1lJ.fl.t-(: Contr·o/) Vo1. AC~21, pp. ;Jl !J-:l:1S.
Zhang, C., (19!J2), A dllal rate compensat.or for system loop zero assignment., .'~hr;fnTl.5 and Conho/ Letter's, Vol. 19, pp. 2'li)~2;32. Zhang and V. Bloudel, (1993). SimuIt.aneous st.abilization Ilsiug an LTI c.ornpensat.or ;:uld a sampler and hold, Int.. .1. Control, Vol. &7, No. 2, PI', 29;3 ~ 308. Zhang, .1. and C. Zhang, (H)I)Fi), R.obust.ness analysis of (".(mt.rel syst.em using generalized sample hold functions, P7"(!(:. :1.'11(/ IEEE CDC, Orlaudo, USA. Zhang, .1. alld C. Zhang, (1 m;5), Stflhilit.y margin analysis of disnet.e periodically t.in;e varying controllers, PmL ,1411, IEEE CDC', New Orle;ws, USA.
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