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Robust Filtering for Uncertain Delay Systems Under Sampled Measurements
2d-024
Copyright@ 19961FAC 13th Triennial World Congre~s, San Francis{:u, USA
ROBUST FILTERING FOR UNCERTAIN DELAY SYSTEMS UNDER SAMPLED MEASUREMENTS
Peng Shi'
.. Cr!ntre f or Industrial & Applied Mathematic,1f School of Mathematic .• The Univer!~ity of South Au.,tralia
SA 5095, Australia mat..,[email protected]".uni..m.cd1L.au
Abstract. This paper is concerned with the problem of robust. H:x;,
filt~l"illg
for a
cla.."is of systems with parametrk ll!\<: ert.ainti~s and unknown time delays uuder sam· pled mea.
sampled mea.'mrt".l Uents. which would guarantee a prescribed Hoo performance in the continuous-time context, irresp~tjve of t.he parametf"r nn('..ertainties and unknown timc-dday~. Both t.he cases of finit.e and infinit.e hori7.on filtering: are studind. It has heen shown t.hat the above robust. B oo fillcring prohlem can be solved in terms of differential Riceat.i equations with finit,e discrete jumps. Keywords. time-delay systems,
lln c~rt.ai n
syst.ems, sampled mea."illrements.
1. INTRODUCTION Control of time-delay syst.ems is a subject of great practical importa.nce which ha.;; attracted a lot of int.f~rest for several decades; see e.g. Malek-Zavarei and .1a.mshidi (1987). R ecently! some rf'~<;eardl work ha...:;: been conc~m trated on the design of robust control for uncertain systems with time-delay, III the work of Thowsen (1983), nonlinf'.ar t.ime--delay state feedba.c.k cont.rollers have been considered whereas Ha..;;anul et al. ( 1986) and Shen et ,1,1. (1991) ha.ve fOCllsp.d on memorylffi.o;; linf>.a.r state feedback control. In the meant.ime, mudl attent.ion has h~n paiet t.o the prohlem of HOC) filt.ering, and a lot of progress hac; been made. sp.e, e.g .. Grimble allo Sayeo (1990), Bernstein and Haddad (1989), "fagpal and Khargonekar (1991) and t he references therein. I\·feanwhile, the H oo filtering problem for system with parameter 1Incert.ainty ha" also been studied by Xic et at. (1991) , whidl address the design of filte-..rs guaranteeing both the robust
3081
st.ability and a. prescribeci Hoo performance for the filtering error dynamics in the face of parameter WlcertrLinty. Pllrt.Ilf'!rmore, Hoo filte ring and control for linear sampled-data systE".ffi ha~ been tar.klp.d by SlUl et al. (1993). The desib'11 of robust filters for uncertain lineal' continuous-t.ime systems 1Isin2; sampled measurements has been investigated by Shi e.: al. (1993). In t.his p..per, we invp.stigate aD H oo filtering problem for a da..<;s of lmcertain continllolls-time systems under sampled measurements. The class of uncertain systems is described by state space model wit.h real time-varying norm-bOlluded parameter Imcertaint.ies, 11Ilknown timedela.ys in the state equation. Here attention is fO(,;lIsed on the design of linear dynamic 111 t,ers using'local sampled mea")urements which guar.'1.ntee the robust. stability as well as a. prescribed H oc performance for the filtering f'..I'ror dynamics, irrespect.ive of paramet.er uncertainties and ImknO\\"Il time-delays. The perfonua:uce measure we use is defined directly in the continuous-time cont.ext.. It
IIF(t)11 ~ 1, "VI ~ 0,
is asSllInNJ that the initial state of the uncertain system is unknown and hoth the c.a.';;f!S of finite and infinite horhwn filtering are considered. 'We show that the robust Hoo sampled-data filtering problems can be l"iolved via scaled output feedbac.k Hoo sampled-data control problems that incorporates unknown initial states and involves neither parameter uncertainties nor unknown time-delays. Therefore, existing results 011 Hoo sampleddata cont.rol sll('h a.", those in Kabamba and Hara (1993) and Sun ~t al. (1993) can lw applied to obtain solutions to the problem of robust H 00 sampled-data filtering for the uncertain systf!ms.
IIF,,(t)11 ~ 1, 'Vt ~
°
(6)
with the elements of F and FOI being Lebesgue measurable. Observe that a{x, t, 0) can be expressed as a linf'..ar transformat.ion of x(t), i.e.
n(x,t,O) = Wx(t)
(7)
where W is a known real constant matrix representing row permutations of x(t). Assumption 1 11<>(x,t,x)ll[o,T] ~ IIWx(t)ll[o.T], 'Vt,r
Remark 2 If. .
(Ed: .;:(t)
= [A + Ll.A(t)]x(t) + [A. + Ll.A.(t)],,(x, t, T) +Bw(t), t E [0.1'], x(O)
z(t)
= L.r(t).
t
y(ih) = Cx(ih)
="0
E [0.1']
+ Dv(ih),
(1)
(2) ih E (0,1')
(3)
many stat~ <~pace modeL~ of delay systems and can be v..~ed to n~pn~.'lent many important phy.'lical . sy8tem'l (.'lee, Malek-Za"Jarei and JamfJhidi (1987) and the references therf.inj. Al.'O note that the uncertain matrices F and Fa aTP- allo1JJt:d 1.0 be state-dependent, i.e. F = F[t,x(t)] and Fo = F.[t,x(t)] as long as (5)(6) are .,ati.'fied. In this paper W~ arc conc~rncd with obtaining an estimat.e Zc (t) of z (t) over a horizon [0 1 T) via a linear causal filter F for (1)-(3) using t.he measurements {y(kh), 0 < kh ~ t}, and where no a priori estimate of the initial state of (1)-(3) are assumed. The filter is required t.o provide lmiformly small estimate error, e(t) z{t) z,(t), t E [0, T] for any w E L,[O, 1'], 1) E 1,(0,1'), Xo E ~Rn and for all admissible uneertainties and unknown time-delays. More spedfic.a11y, the robust H= sampleddata filt.eriug problem we address is as follows: Given a M:alar , > 0 and an initial state weighting matrix R = RT > 0, de.~ign a liner.,r cau.'>al filter F ba<,>ed on the .'>l1.mpled measuTe1wml ... , y(ih), tmch that the filtering error dynamics
=
a(x, t, T) =
(I - Te 1) x,,(1 - Te 2 ) ... x'n" (t - xcn.J]T ikE[I,2, ... ,n], rkE[I,2, ... ,o]' k=I,2, ... ,n o (4) [X;l
=
In syst.em (EJ), we also need to define x(t) 0, t E [-Tmax,Oj, where Tmaa::= max{7i}, i = 1,2, ... ,701 , In the above, x(t) E iRn is the st.ate, Xo is an initial state, wet) E RP is t.he process noise, y E ~Rm is the sampled mea.",llrement, v E Rq is the measurement noise, Z E Rr is the linear combination of t.he st.ate variable t.o be estimated, 0 < hEiR is the sampling period, i is a posit.ive integer, 7 is a vector of unknowIl but bounded time delays of the system, A, Aa, E, C, D and L are known real time-varying bounded matrices of appropriate dimensions that d~sc.ribe the nominal syst.em wit.h A, Act, B and L bf"ing piecewise continuous and ~A and .6. Ao are real time-varying matrix functions representing norm-hounded parameter uncertainties. The admissible parameter lUlcertainties are assmned t.o he modeled as
• In the finite horizon ca ... e,
IIz - z,l!Jo.T] < I
, {2 2 }1/2 (9) IIwlho.Tl + Ilvll(o,T) + Xo R.~o T
fox any non-zero (w, 1), xo) E L2[0, 1']EBI2(0, 1')EB!Jln and for all admi ....'1ible uncertainties and unknown time-delays. • In the infinite horizon c(...~e, the estimation error dynam..c." z(t) - z,(t) i., 91abally uniformly asymptotically stable and (9) holds far all admi., .• ible uncertaintif~$ and unknown time-delaY8 as.T -t 00.
(5 )
3. ROBUST H= FILTERlNG
whf're E E jRiX1'l, Ea E Rj..,xn", HE Rnxi and Ha E Rnxia are known H"..al time-varying matriees heing pieeewise continuou.."" and F E ~Rixj and FOI E ;Ri,,,xj,,, are unknown matrix f1lnctions satisfying
III t.his sectioll WP. will provide solutions t,o the Hoo filtering problems for t.he system (1)-(3) for bot.h the finite and infinite horizon cases. It will be shown that these 3082
problems can be 8olv~d in terms of H 00 control synthe. ",is problems for sampled-data linear systems which do not involve parameter uncertainties and unknown timedelays. "Ve first, consider the problem of robust H oo analysis for a class of uncer t.ain systems with finit.p. discrete jumps.
(E, ) : x(t}
for any non-zero (tv, V,x o) E L.[O, T] (l) I,(O, T) (I) Rn and for all admi.•.• ible uncertainf.if..i and unknoTlm timedelays if there exi,.,t po, 0, 'It E [0, T ), such that.
=
= [A + EiA)x(t} + [Ao + L>A o )a(x, t, r) = Xo
+Bw(t }, I ~ ih, x(o)
i'(t) + AT P + PA + 7 - ' PB(v, '\}B(v, ~)T l'
(to)
x(ih) = Adx(ih- ) + lldV(ih}, ih E (0, T) (ll)
[0. T)
.(1) = L x( I}, lE
+C(V, ,\)TC(v,'\) + L,T L
(12 )
[B
{(ilt) = Ad{(ih -)
i(I}=
1.
+ Bd .;( ih},
[ c(~'\) l E(t).
ih
(18) ( 19)
>0 P( ih) + .1:\, P (ih+}B,
I - Br P(ih+)Bd
. ["t'I PtO}
B;\' P(ih+}Bdr'
Br
(20)
P( ih+)_4.d
< 0 (21)
< "I'R.
(22)
By ~onsid~ring (16)-( 17) and "lsing matrix inequalities in Waog et al (1992), (IB) implies that
P + [A + HFE)Tp + p[A + [{FE) +_1-' P BBT P + P[A o + H . F.E. ) ·[A o + HoFoEo]T P + LT L + WTW < 0,
(13)
I ~ ih(23 )
In view of (10) and (18), it. is . ,,,ily verified that for any rE Uh , ih+h)
ih E (0, Tl{14)
lE [O,T)
2
A;\' P( ih+)Ad -
"lB(v, ,\}]W},
t ~ ih , {to} = (0
t ~
P I T) = 0
where ;r E !Rn is t.he state vector, Xo is an 11nknown initial state, z E RP is the output. tJ) E Jl:9 and tJ E R' are th e dist.urbance inputs, Ad and Bd are known real time-varying bounded matrices, o(x, t, T) is a..;; in (4) and satisfies (7) a.nd a ll the other mat.rices are the same M in (5}-(G). In the sequel we shall establish an interconnection hetween the robu,t H = performance analysis for (1O}-(12) and t.hat, of t,he following auxiliary system
(E~ ): ~(I) = A(I) +
< 0,
(IS )
J• :t • =J
where { E !Rn is the st.ate vect.or, {o is an lUlknown init.ial sta.t.r., rj E :tt: n + q and fj E RI arC! tlw di sturban ce input' whk.h belong to L 2 [0, T) an<1I,(O, T) , resp er.t,ively, E )Rn+p is t.h~ out.put, UI E ~q and v E ~l are t.he disturbance iI~PlltS, A, Ad,_B Bd and L are the same n..o;; in (10)-(12), B(v,,\) and C(v,'\) are define as foll ows:
(x T Px}dl
= .•T P( T)"-( T) -
x T (ih+ }P(ih+)x(ih+ )
ih+
z
","M(I}xdt + "I'lIwlll'h,,1
j
- [l zII1'h.') -llrll['h"J
ih+
-1I1'II[ih .•) + lI"IIi'",.) -IIWxll!,h,.) -
-
T
I
2
.,.
T
R(v,,\ )R(v, ,\) =,HH +A.,,(I-,\E" E o
,,-
I
)- 1A"T
T
+ A2H«H(Jf
where M(t) denot.es the left. hard ,ide of (18), i.e . .'11(1 ) < 0, VI E (i h , ilt + h) and
(1G)
C(v, '\f C( v, '\) = ," ETE+ WTW
= a(x, I, T) - (A. + L>Ao)T P x ,. = "1['" - "1- ' B T PT.]
I'
(17)
Whf>I P. Aa is t\.'5 in (1), E ~ E(Ir , H and Ho are thesamea..<1i in (5), .", is a..~ in (7) and v a.nd ). are positiv(! sc.aling paJ'anwtcrs FOndl t.hat J - ).2 E(O > O. First, we derive the following Hx, p~rforma.nt.p. analysis r E'~'i ll1 t for ( l O)-{ 12) over finite horizon.
Al.;;o c.onsi.ring (11) and by (",{·mplet ing the sqnares IISing ( 20) and (21) , we have t.ha';
e;;
x T Px [:~+
2
> 0.,
2
= xT(ih+ }P(ih+}x (ih+} -
xT{ih-)P(ih)x(ih-}
= xT{ih-)Q(ih).T(ih-) + -? III{ih}II' -1I .. (ih)II'.
Theorem 3 Co nsider the "y"lem. of (10)-{12) .• ati,IYing (S)-(Ii) and (7). Gi"en a ."alar l' > 0 and an inilial ,.tate weighting matrix R = RT
(24)
where Q(ih} stand. for the left. hllml ,iue of (21), i.e. Q(ih) < 0 and
the ,
T} ./,
11.lho.TJ < l' {Ilwll[o.TJ + IIvlll o.T) + a,oRxo
3083
(25)
hI -
{v(ih) -
BT P(ih+)Bdt 1 BT P(ih+)AdX(ill -)} .
W~ hil.ve that
x TM xdt
+
L
(31)
d~1ine
., T(ih-)Q(ih)x(ih- )
dated wit,h the unforced system (10) (setting wIt) '" 0):
i h E( O,T )
+, 'lIwll[o,T[ -lI zll[o ,T] + ,'lIvlli•.n -lIrll[o 'l1 -1I1111[o,T) + lI all[0,7) -IIWxll[o ,T) - IIfllIlo.T) < " [1I'"II[o,T) + II vllionj-lI zll[o,Tj +[[all[o,Tj - [[W xlllo,T] (26) for any non-zero (w. v, IO) E £, [0. T] e I, [(0, T) e ~n. By taking into aCCollnt (8) and the fact that P(T) and prO) < ,'R, (26) implies that
r
Vex , !) = ",T(t)Px(t)
.,[x(s) , .s, T]T o [x(s), s, Tlds.
o
=0
:-Iote that in view of (7) and P(t) > 0, t E [0,(0), V(x, t) > 0 whenever x(t) # O. Furthermore, along any state trajectory of the system of (10) we have that for all t E (ih,ih+h)
~V x 1 _
x(l) dt ( , ) - [ O(X,I, T) ]
.[
.[
<1>(1) , , < -I { [[wll[o,oo) + [[1111(0,00) + X To R.xo }'/'
x(t)
o:(x , t , r)
+ HoFoEo)]
P
-I
]
(32)
= P + [A + HF(I)Ef P + PIA + HF(t)E] + WTw.
LlV[:r(ih)1
PROOF. Similar to t.he proof of Throrp.1n 3, using Theorem 3.2 in Shi et al. (1992) , it. follows t.hat there exists
= V[x(ih+), ih+l- V[ x(ih-) ,ih-]
=x"(iI,-) {}t~P(ih + )Ad + A~P(ih+)Bd h' J - BI P(ih+ )Bd[-l .BI P(i"+ )Ad
O. 'It E [0,00) s"ch
-P(ih; }x(i h- ) Thcrpiore 1
P + [A + HFEfp + P[,4,+ HFE] + , - 'PBB'I'P
WP.
+ H.F.E.]·[A. + HoF. Eo t P +£T £ + WTW < O. t # il> (27)
< O.
conclude from the above that the r.qui-
librium state x
+P[A.
,'I - BTP(ih+ )Bd > 0
PtA.
T
Xext. in view of (2i) it. follow , immediat.ly from (32) t.hat dV~~,t) < 0 whenever x(t) # 0 fo r all! # ih. On the other hand l by conside::ing the unforr.en system (11) (set.ting vUh) '" 0) and (31l) we obt ain that. for any non-zero ;r,(ih-)
j()r any non- Zf:.m (lV, v, xo) E L:dO) 00) ffi 12 (0. 00) ffi lRl1 and for all adtnl~~!;ible uncertaintif;.t; and u.nknown tim edelflY.., t/ th" .~y.dp.'m (2.V i..'1 exponentiaJly stable and there exi.,t po.~ i tivp. M:aling parameter,t; v and), .mr.h th.at >.2 E7: E~ < J and J (I:, . R. oo) < "{.
ArP(ih+ )Ad - P(ih) + A:\, P(ih+)Bd
T
where
.'HLti,'ifi r. ,~
P( oo ) =0
<1>(t )
+ H.F.E.)
(.4.
Theorem 4 Consider the system of (lO}-(1 2) satisfy· ing (5) -(6) and (7). Given" .muar -; > 0 "nd an initial ., tate weighting matrix R = RT > 0, the .o;y"tr:m i.'i globally uniformly a..lOymptot;'cally !~table about the Drigin and
= pT(t ) >
o
-J
whenew!I' (111, I) , Xo) 'I 0 ilnd for all admissible unCE"..rtainties and unknown time-delays , which completes the proof. Theorem 3 can be easily extended to the infinite hori:7.0n case A.<:; hdow:
matrix func tion p r' ) tha t
+ ) [W x(sJ[T[Wx(s)[ds
,
[[zll[o,T) < " [II"'[[[O,TI + 1I 1' [[lo,T) + ·,r Rxo j
[[ z [lro , ~)
(30)
ble uncertainties and unknown time-delays. To t.his cnd, the following piecewise .:ontinuous function 3550-
T
o
<0
First, we need to ""sert that the system (10)-(12) is globally uniformly asymptot.ically stable for all admissi-
x T(T)P(T).r.(T) - x T (O)P(O)x(O)
J
BT P(ih+)Bdj - l BT P(ih+).4d
PtO) < ..,' R.
Dy combining (24) and (25) over all possible ih in [O.T],
=
. [-y'I -
= 0 of the sys :em (10)-(12) is globally
uniformly a.<;ymptotically stable for all admissible uncert.aint.ies and unknown time-delays, Finally, the proof of that.
(28) (29)
[[ z IlIO,~) 3084
<,
{
,
[[1/)1I[0, ~)
, + [[1·1[(0,00) + XoT R xo}' / '
Then, given a lIcalar 'Y > 0 and an initial statc lIJCJght= RT > 0, the m~",4t Hoo filtering problem as.• ocinted with (1)-(3) is solvcble via the linear filter F over the horizon [0, T] if then exi... t po"titi'IJe "caling parameter,,> 1/ and .\ such that
for any non-zero (w, V, .TO) E L, [0, 00) $ 1,(0,00) 6l !Rn and for all a.dmissible uncertainties and unknown timedelays is almost identical to that of Theorem 3, p-xcept t.hat. T .... 00. Xow we sho.." that the problem of robust. Hoo sampleddat.a filtering of uncertain time-delay system (1 )-(3) can be converted into scaled H r= sampled-data control problem for linear system which does not involve any of parametcif'. llncf!rt.aiut.y ami nnknown time-delays. To t.hi~ end. associated wit,h system (1)-(3), we introduce the following auxili.vy sys tem:
(Efl: x c(t) = A.xAt) + x,(O) =
z,( t)
where
Xc
rB t
ft e
1'·
PROOF. The cont.rolla", n , = Fy, (ih) for system (33)(35) is given by
XcO
8,(;11 )
{(ih) = . , (t)
A, B, C, D, E and L are !.l,e same as in ( 1)-(3), W is as in (7), and 8(v, A) LS
(.onr.folled outPUf., the mat.rice>
t E [O,T]
((I) =
with Aa. Eo, H , Ho as given in (1) and (5), v and A b€,jng positive scaling para.meters s11ch that. ,\2 Eo < I. The linear filt.er we consider for (1 )-(3) is of the form:
Er
= A , I1,(t), top i/o,
(1, (0)
=
°
(37)
" ,(i/O ) = .4. d l1,(ih- ) + Bdy(ih) , ih E (O,T) (38 ) .,(1) = L , I1, (t) , lE [O,T]
(41)
=
[B, ,),8,] III,(t) t ¥- ih, {to) = [x,(of W Ad{(ih - } + 8dl ,,(ih), ih E (0, T)
(44)
[g:] ((I).
(45)
t EO, T]
(43)
where
[I-;;H
(F) : ,;-,(t)
(40) (42)
defined as follows:
_
°
ih E (O,T)
~(t)=.4:{(t)+
E )RQ+m+q
and Vc E Rq are the disturbance inputs which belong to L,[O , T] and f,( O, T ), respect.ively, y, E 'J1' is the measured output, ~I > 0 is t.he desired rlist.nrbance at.teOllation level for system of (1)-(3), .,(t) E !Rn +p+ ~ is the
B(V,A ) =
=
Now, the dosed-loop system of (33}-(35) alld (40)-(42) is given by
ih E (0,T)(35) tV e
8,(0)
= AdO,(ih- ) + E'dy(ih),
= L , O,(t),
z,(t)
(34)
is an unknown ini-
E 1RP is the control inpllt,
= A ,O,(t), t ¥- ih,
Ii,( t) (33)
= Cx,(ih) + Dv,(ih),
E !R11 is the st.ate, and
(i) )., E,!: Eo < I; (ii) With contTol law .. ,(t) = -'Ye f OT (.93)-(95) the resulting clo." d loop .• ystem, (E"I , .• ati"fie .• J (E" , R , T) <
7B(v, AlIIIIc(t}
¥- ih
= [ ~ ]",(1) + [ ~I] ",(I), t E [O,T] y, (i")
tial statf'!,
XcQ,
ing matrix R
[x~(t) 8~(tW,
Ba=[B~D],Cl=[L
~p
-L, ],
+B,III(I), '1(ih) = A d '1(i/O-)
= C''I(t} ,
eft)
lA .• + if. FoE. ]a(I), t , 7) t ¥- ih, "~to} = [",T(O) O]T (49)
+ BdV(ilo) ,
iloE(O,T)
t E [0, T]
(50)
(51)
where
H
3085
C2 =[~~~l(48)
Jj(t) = [_4 + H FE]I)(t ) +
11;(tW, ';('1,t, T) = "'( 'I. t, T) (52)
= [~: ] , H. = [ ±:~" E = [v 0J, Ea = AE"
Theorem 5 Consider the .,y.,tem (1)-(.?) .• ati.•fying (5)(6) and (7) and let (F) denotes a lineaT timc-var,ying jilr.t'- r wi.th zero initifll condition of the form (87)-(.99) .
(46 )
model:
'1ft) = [xT(t)
unc.p.rtain sy"t.p.m (1 )-(3) on finite horizon.
1J
where B(l', A) is as in (36). ::\P..xt, the filtering error, e = z· - Zi' a.ssociated with system (1)-(3) and fiUer (37)-(39) is given by the following star..e
(37}-(39). 0111' first result dealR wit.h the robust. H'XJ filtering of
[~
_4d = [B:C L], B, = [~] , B, = [B(~').)l (47)
(39)
where the dimension of filter , ne. and the matric.cs Ad. A c, B d and L , are to he chosen. :-.fot.e that (37)-(39) can be interpreted a.~ linear discrete time varying filt.er (38) followeO by int.erpolation funr:t.ion (37). which generates t.h~ sigual ~t.iIIlates between the sampling instant.s. Digital fi1t,~l's will! a 'l.t"ro-order hold are special cases of
A=
],
A"
= [~o 1(53) (54)
Note that. for any'! E
~Rn+T\~
be converted to related Hoo sampled-data control problems without paramet.er 1lncert.ainties and unknown t.imedelays, which can be solved in terms of differential Ric· cati equations with finite discr,~te jumps.
1I<'>('1,t,O)1I = 11<>('1,t,O)11 = IIW'1(t)11 where IV = [W OJ, Now, using (46)-(48) and (52)-(54), it is easy to sce that
Referenc:es
BJf; = f[f[T + .4:. (I - E~E") - I .4:~ + f["f[~ (55) cic,=£TE+WTW (56)
Bernstein, D. S, and Haddad, W, M, 1989. St.eady-state Kalman filtering with an H(
Finally, by considering the systems (43)-(45) and (49)(51), the identities (55)-(56) and condition (ii) of the theorem statement l t.he desired result. follows immediately from Theorem .'3. The next theorem deals with the problem of robust Hoo filt.ering of ntlc.ertain system (1)-(3) on infinite horizon. Theorem 6 Con"id" the ,'y"tem (1)-(3) .. ati,'fying (5)(6) and (7) and let (.r) denote" a linear time-varying filter with zero initial condition of the form (S7)-(39)' Then, given a .~calar . . ,. > 0 and an initial ,~tate weighting matri.7: R = RT > 0, the mbwJt Hoc. filtering problem a,~!wciated with (1)- (.'1) -i.~ solvable ?Jia the linear filter :F over the horizon [0,00) if there exi!~t positive scaling parameter,~ v
and ,\ fI'Il.ch that
(i) ),2E;;E. < I; (ii) With the control law ",(t) = .ry, for (3,1)-(35) the re,mltmg do,~ed loop fty.r;tem, ('£.d), i... exponentially I;;tahIe and ,~ati.~fie.~ J(Lcl,R,oo) < ''t. PROOF.Along t.he same line
a.-<;
in the proof of Theorem
5, it. can be established by applying Theorem 4 to the
closed-loop system of (33)-(35) with (.r), and t.he do"dloop system of (1)-(3) wit.h (.r),
Remark 7 It is shown hy Theor€m.r; 5 and 6 that the problem of rohu8t Hoo sampleAt-data filtering for uncerta'in <~y8terns of the Innn (1)-(3) can be cnn?Jerted intn scaled Hoo output feedback control problem with ,'1ampled mea.'1u.rem.eT/,t.~ which in1Jo/1Je.r; neither pI!mmeter ?Lncertaintie.r; nor unknown time-delaYfJ. Therefore, existing results on Hoo 8ampled-data control ,'Ouch as tho,<;e in Kabamba and Hflm (199,9) and Sun et aL (1993) can he 1Uu~d to de,r;ign robust Hex> jilter,~ for uncertain ,r;ampleddata !ly,<;tem.<; of the form. (1)- (.9). VVhen there i.~ no timedelay" in (1)-(.1), i,e, a(x,t,T) '" 0, the re"ult" of Theorems 5 and 6 will ,'educe to tho,," of Shi et. al, (1993).
4, CONCLUSION The design of robust filters for a class of continuous-time syst.ems with sampled measurements which are subject to norm-bounoeo paramet.er uncert.aint.ies anollnknown time~delays has been considpxed in this paper. The results obtained have shown that. the above problems can 3086
Copyright@ 19961FAC 13th Triennial World Congre~s, San Francis{:u, USA
ROBUST FILTERING FOR UNCERTAIN DELAY SYSTEMS UNDER SAMPLED MEASUREMENTS
Peng Shi'
.. Cr!ntre f or Industrial & Applied Mathematic,1f School of Mathematic .• The Univer!~ity of South Au.,tralia
SA 5095, Australia mat..,[email protected]".uni..m.cd1L.au
Abstract. This paper is concerned with the problem of robust. H:x;,
filt~l"illg
for a
cla.."is of systems with parametrk ll!\<: ert.ainti~s and unknown time delays uuder sam· pled mea.
sampled mea.'mrt".l Uents. which would guarantee a prescribed Hoo performance in the continuous-time context, irresp~tjve of t.he parametf"r nn('..ertainties and unknown timc-dday~. Both t.he cases of finit.e and infinit.e hori7.on filtering: are studind. It has heen shown t.hat the above robust. B oo fillcring prohlem can be solved in terms of differential Riceat.i equations with finit,e discrete jumps. Keywords. time-delay systems,
lln c~rt.ai n
syst.ems, sampled mea."illrements.
1. INTRODUCTION Control of time-delay syst.ems is a subject of great practical importa.nce which ha.;; attracted a lot of int.f~rest for several decades; see e.g. Malek-Zavarei and .1a.mshidi (1987). R ecently! some rf'~<;eardl work ha...:;: been conc~m trated on the design of robust control for uncertain systems with time-delay, III the work of Thowsen (1983), nonlinf'.ar t.ime--delay state feedba.c.k cont.rollers have been considered whereas Ha..;;anul et al. ( 1986) and Shen et ,1,1. (1991) ha.ve fOCllsp.d on memorylffi.o;; linf>.a.r state feedback control. In the meant.ime, mudl attent.ion has h~n paiet t.o the prohlem of HOC) filt.ering, and a lot of progress hac; been made. sp.e, e.g .. Grimble allo Sayeo (1990), Bernstein and Haddad (1989), "fagpal and Khargonekar (1991) and t he references therein. I\·feanwhile, the H oo filtering problem for system with parameter 1Incert.ainty ha" also been studied by Xic et at. (1991) , whidl address the design of filte-..rs guaranteeing both the robust
3081
st.ability and a. prescribeci Hoo performance for the filtering error dynamics in the face of parameter WlcertrLinty. Pllrt.Ilf'!rmore, Hoo filte ring and control for linear sampled-data systE".ffi ha~ been tar.klp.d by SlUl et al. (1993). The desib'11 of robust filters for uncertain lineal' continuous-t.ime systems 1Isin2; sampled measurements has been investigated by Shi e.: al. (1993). In t.his p..per, we invp.stigate aD H oo filtering problem for a da..<;s of lmcertain continllolls-time systems under sampled measurements. The class of uncertain systems is described by state space model wit.h real time-varying norm-bOlluded parameter Imcertaint.ies, 11Ilknown timedela.ys in the state equation. Here attention is fO(,;lIsed on the design of linear dynamic 111 t,ers using'local sampled mea")urements which guar.'1.ntee the robust. stability as well as a. prescribed H oc performance for the filtering f'..I'ror dynamics, irrespect.ive of paramet.er uncertainties and ImknO\\"Il time-delays. The perfonua:uce measure we use is defined directly in the continuous-time cont.ext.. It
IIF(t)11 ~ 1, "VI ~ 0,
is asSllInNJ that the initial state of the uncertain system is unknown and hoth the c.a.';;f!S of finite and infinite horhwn filtering are considered. 'We show that the robust Hoo sampled-data filtering problems can be l"iolved via scaled output feedbac.k Hoo sampled-data control problems that incorporates unknown initial states and involves neither parameter uncertainties nor unknown time-delays. Therefore, existing results 011 Hoo sampleddata cont.rol sll('h a.", those in Kabamba and Hara (1993) and Sun ~t al. (1993) can lw applied to obtain solutions to the problem of robust H 00 sampled-data filtering for the uncertain systf!ms.
IIF,,(t)11 ~ 1, 'Vt ~
°
(6)
with the elements of F and FOI being Lebesgue measurable. Observe that a{x, t, 0) can be expressed as a linf'..ar transformat.ion of x(t), i.e.
n(x,t,O) = Wx(t)
(7)
where W is a known real constant matrix representing row permutations of x(t). Assumption 1 11<>(x,t,x)ll[o,T] ~ IIWx(t)ll[o.T], 'Vt,r
Remark 2 If. .
(Ed: .;:(t)
= [A + Ll.A(t)]x(t) + [A. + Ll.A.(t)],,(x, t, T) +Bw(t), t E [0.1'], x(O)
z(t)
= L.r(t).
t
y(ih) = Cx(ih)
="0
E [0.1']
+ Dv(ih),
(1)
(2) ih E (0,1')
(3)
many stat~ <~pace modeL~ of delay systems and can be v..~ed to n~pn~.'lent many important phy.'lical .
=
a(x, t, T) =
(I - Te 1) x,,(1 - Te 2 ) ... x'n" (t - xcn.J]T ikE[I,2, ... ,n], rkE[I,2, ... ,o]' k=I,2, ... ,n o (4) [X;l
=
In syst.em (EJ), we also need to define x(t) 0, t E [-Tmax,Oj, where Tmaa::= max{7i}, i = 1,2, ... ,701 , In the above, x(t) E iRn is the st.ate, Xo is an initial state, wet) E RP is t.he process noise, y E ~Rm is the sampled mea.",llrement, v E Rq is the measurement noise, Z E Rr is the linear combination of t.he st.ate variable t.o be estimated, 0 < hEiR is the sampling period, i is a posit.ive integer, 7 is a vector of unknowIl but bounded time delays of the system, A, Aa, E, C, D and L are known real time-varying bounded matrices of appropriate dimensions that d~sc.ribe the nominal syst.em wit.h A, Act, B and L bf"ing piecewise continuous and ~A and .6. Ao are real time-varying matrix functions representing norm-hounded parameter uncertainties. The admissible parameter lUlcertainties are assmned t.o he modeled as
• In the finite horizon ca ... e,
IIz - z,l!Jo.T] < I
, {2 2 }1/2 (9) IIwlho.Tl + Ilvll(o,T) + Xo R.~o T
fox any non-zero (w, 1), xo) E L2[0, 1']EBI2(0, 1')EB!Jln and for all admi ....'1ible uncertainties and unknown time-delays. • In the infinite horizon c(...~e, the estimation error dynam..c." z(t) - z,(t) i., 91abally uniformly asymptotically stable and (9) holds far all admi., .• ible uncertaintif~$ and unknown time-delaY8 as.T -t 00.
(5 )
3. ROBUST H= FILTERlNG
whf're E E jRiX1'l, Ea E Rj..,xn", HE Rnxi and Ha E Rnxia are known H"..al time-varying matriees heing pieeewise continuou.."" and F E ~Rixj and FOI E ;Ri,,,xj,,, are unknown matrix f1lnctions satisfying
III t.his sectioll WP. will provide solutions t,o the Hoo filtering problems for t.he system (1)-(3) for bot.h the finite and infinite horizon cases. It will be shown that these 3082
problems can be 8olv~d in terms of H 00 control synthe. ",is problems for sampled-data linear systems which do not involve parameter uncertainties and unknown timedelays. "Ve first, consider the problem of robust H oo analysis for a class of uncer t.ain systems with finit.p. discrete jumps.
(E, ) : x(t}
for any non-zero (tv, V,x o) E L.[O, T] (l) I,(O, T) (I) Rn and for all admi.•.• ible uncertainf.if..i and unknoTlm timedelays if there exi,.,t po,
=
= [A + EiA)x(t} + [Ao + L>A o )a(x, t, r) = Xo
+Bw(t }, I ~ ih, x(o)
i'(t) + AT P + PA + 7 - ' PB(v, '\}B(v, ~)T l'
(to)
x(ih) = Adx(ih- ) + lldV(ih}, ih E (0, T) (ll)
[0. T)
.(1) = L x( I}, lE
+C(V, ,\)TC(v,'\) + L,T L
(12 )
[B
{(ilt) = Ad{(ih -)
i(I}=
1.
+ Bd .;( ih},
[ c(~'\) l E(t).
ih
(18) ( 19)
>0 P( ih) + .1:\, P (ih+}B,
I - Br P(ih+)Bd
. ["t'I PtO}
B;\' P(ih+}Bdr'
Br
(20)
P( ih+)_4.d
< 0 (21)
< "I'R.
(22)
By ~onsid~ring (16)-( 17) and "lsing matrix inequalities in Waog et al (1992), (IB) implies that
P + [A + HFE)Tp + p[A + [{FE) +_1-' P BBT P + P[A o + H . F.E. ) ·[A o + HoFoEo]T P + LT L + WTW < 0,
(13)
I ~ ih(23 )
In view of (10) and (18), it. is . ,,,ily verified that for any rE Uh , ih+h)
ih E (0, Tl{14)
lE [O,T)
2
A;\' P( ih+)Ad -
"lB(v, ,\}]W},
t ~ ih , {to} = (0
t ~
P I T) = 0
where ;r E !Rn is t.he state vector, Xo is an 11nknown initial state, z E RP is the output. tJ) E Jl:9 and tJ E R' are th e dist.urbance inputs, Ad and Bd are known real time-varying bounded matrices, o(x, t, T) is a..;; in (4) and satisfies (7) a.nd a ll the other mat.rices are the same M in (5}-(G). In the sequel we shall establish an interconnection hetween the robu,t H = performance analysis for (1O}-(12) and t.hat, of t,he following auxiliary system
(E~ ): ~(I) = A(I) +
< 0,
(IS )
J• :t • =J
where { E !Rn is the st.ate vect.or, {o is an lUlknown init.ial sta.t.r., rj E :tt: n + q and fj E RI arC! tlw di sturban ce input' whk.h belong to L 2 [0, T) an<1I,(O, T) , resp er.t,ively, E )Rn+p is t.h~ out.put, UI E ~q and v E ~l are t.he disturbance iI~PlltS, A, Ad,_B Bd and L are the same n..o;; in (10)-(12), B(v,,\) and C(v,'\) are define as foll ows:
(x T Px}dl
= .•T P( T)"-( T) -
x T (ih+ }P(ih+)x(ih+ )
ih+
z
","M(I}xdt + "I'lIwlll'h,,1
j
- [l zII1'h.') -llrll['h"J
ih+
-1I1'II[ih .•) + lI"IIi'",.) -IIWxll!,h,.) -
-
T
I
2
.,.
T
R(v,,\ )R(v, ,\) =,HH +A.,,(I-,\E" E o
,,-
I
)- 1A"T
T
+ A2H«H(Jf
where M(t) denot.es the left. hard ,ide of (18), i.e . .'11(1 ) < 0, VI E (i h , ilt + h) and
(1G)
C(v, '\f C( v, '\) = ," ETE+ WTW
= a(x, I, T) - (A. + L>Ao)T P x ,. = "1['" - "1- ' B T PT.]
I'
(17)
Whf>I P. Aa is t\.'5 in (1), E ~ E(Ir , H and Ho are thesamea..<1i in (5), .", is a..~ in (7) and v a.nd ). are positiv(! sc.aling paJ'anwtcrs FOndl t.hat J - ).2 E(O > O. First, we derive the following Hx, p~rforma.nt.p. analysis r E'~'i ll1 t for ( l O)-{ 12) over finite horizon.
Al.;;o c.onsi
e;;
x T Px [:~+
2
> 0.,
2
= xT(ih+ }P(ih+}x (ih+} -
xT{ih-)P(ih)x(ih-}
= xT{ih-)Q(ih).T(ih-) + -? III{ih}II' -1I .. (ih)II'.
Theorem 3 Co nsider the "y"lem. of (10)-{12) .• ati,IYing (S)-(Ii) and (7). Gi"en a ."alar l' > 0 and an inilial ,.tate weighting matrix R = RT
(24)
where Q(ih} stand. for the left. hllml ,iue of (21), i.e. Q(ih) < 0 and
the ,
T} ./,
11.lho.TJ < l' {Ilwll[o.TJ + IIvlll o.T) + a,oRxo
3083
(25)
hI -
{v(ih) -
BT P(ih+)Bdt 1 BT P(ih+)AdX(ill -)} .
W~ hil.ve that
x TM xdt
+
L
(31)
d~1ine
., T(ih-)Q(ih)x(ih- )
dated wit,h the unforced system (10) (setting wIt) '" 0):
i h E( O,T )
+, 'lIwll[o,T[ -lI zll[o ,T] + ,'lIvlli•.n -lIrll[o 'l1 -1I1111[o,T) + lI all[0,7) -IIWxll[o ,T) - IIfllIlo.T) < " [1I'"II[o,T) + II vllionj-lI zll[o,Tj +[[all[o,Tj - [[W xlllo,T] (26) for any non-zero (w. v, IO) E £, [0. T] e I, [(0, T) e ~n. By taking into aCCollnt (8) and the fact that P(T) and prO) < ,'R, (26) implies that
r
Vex , !) = ",T(t)Px(t)
.,[x(s) , .s, T]T o [x(s), s, Tlds.
o
=0
:-Iote that in view of (7) and P(t) > 0, t E [0,(0), V(x, t) > 0 whenever x(t) # O. Furthermore, along any state trajectory of the system of (10) we have that for all t E (ih,ih+h)
~V x 1 _
x(l) dt ( , ) - [ O(X,I, T) ]
.[
.[
<1>(1) , , < -I { [[wll[o,oo) + [[1111(0,00) + X To R.xo }'/'
x(t)
o:(x , t , r)
+ HoFoEo)]
P
-I
]
(32)
= P + [A + HF(I)Ef P + PIA + HF(t)E] + WTw.
LlV[:r(ih)1
PROOF. Similar to t.he proof of Throrp.1n 3, using Theorem 3.2 in Shi et al. (1992) , it. follows t.hat there exists
= V[x(ih+), ih+l- V[ x(ih-) ,ih-]
=x"(iI,-) {}t~P(ih + )Ad + A~P(ih+)Bd h' J - BI P(ih+ )Bd[-l .BI P(i"+ )Ad
O. 'It E [0,00) s"ch
-P(ih; }x(i h- ) Thcrpiore 1
P + [A + HFEfp + P[,4,+ HFE] + , - 'PBB'I'P
WP.
+ H.F.E.]·[A. + HoF. Eo t P +£T £ + WTW < O. t # il> (27)
< O.
conclude from the above that the r.qui-
librium state x
+P[A.
,'I - BTP(ih+ )Bd > 0
PtA.
T
Xext. in view of (2i) it. follow , immediat.ly from (32) t.hat dV~~,t) < 0 whenever x(t) # 0 fo r all! # ih. On the other hand l by conside::ing the unforr.en system (11) (set.ting vUh) '" 0) and (31l) we obt ain that. for any non-zero ;r,(ih-)
j()r any non- Zf:.m (lV, v, xo) E L:dO) 00) ffi 12 (0. 00) ffi lRl1 and for all adtnl~~!;ible uncertaintif;.t; and u.nknown tim edelflY.., t/ th" .~y.dp.'m (2.V i..'1 exponentiaJly stable and there exi.,t po.~ i tivp. M:aling parameter,t; v and), .mr.h th.at >.2 E7: E~ < J and J (I:, . R. oo) < "{.
ArP(ih+ )Ad - P(ih) + A:\, P(ih+)Bd
T
where
.'HLti,'ifi r. ,~
P( oo ) =0
<1>(t )
+ H.F.E.)
(.4.
Theorem 4 Consider the system of (lO}-(1 2) satisfy· ing (5) -(6) and (7). Given" .muar -; > 0 "nd an initial ., tate weighting matrix R = RT > 0, the .o;y"tr:m i.'i globally uniformly a..lOymptot;'cally !~table about the Drigin and
= pT(t ) >
o
-J
whenew!I' (111, I) , Xo) 'I 0 ilnd for all admissible unCE"..rtainties and unknown time-delays , which completes the proof. Theorem 3 can be easily extended to the infinite hori:7.0n case A.<:; hdow:
matrix func tion p r' ) tha t
+ ) [W x(sJ[T[Wx(s)[ds
,
[[zll[o,T) < " [II"'[[[O,TI + 1I 1' [[lo,T) + ·,r Rxo j
[[ z [lro , ~)
(30)
ble uncertainties and unknown time-delays. To t.his cnd, the following piecewise .:ontinuous function 3550-
T
o
<0
First, we need to ""sert that the system (10)-(12) is globally uniformly asymptot.ically stable for all admissi-
x T(T)P(T).r.(T) - x T (O)P(O)x(O)
J
BT P(ih+)Bdj - l BT P(ih+).4d
PtO) < ..,' R.
Dy combining (24) and (25) over all possible ih in [O.T],
=
. [-y'I -
= 0 of the sys :em (10)-(12) is globally
uniformly a.<;ymptotically stable for all admissible uncert.aint.ies and unknown time-delays, Finally, the proof of that.
(28) (29)
[[ z IlIO,~) 3084
<,
{
,
[[1/)1I[0, ~)
, + [[1·1[(0,00) + XoT R xo}' / '
Then, given a lIcalar 'Y > 0 and an initial statc lIJCJght= RT > 0, the m~",4t Hoo filtering problem as.• ocinted with (1)-(3) is solvcble via the linear filter F over the horizon [0, T] if then exi... t po"titi'IJe "caling parameter,,> 1/ and .\ such that
for any non-zero (w, V, .TO) E L, [0, 00) $ 1,(0,00) 6l !Rn and for all a.dmissible uncertainties and unknown timedelays is almost identical to that of Theorem 3, p-xcept t.hat. T .... 00. Xow we sho.." that the problem of robust. Hoo sampleddat.a filtering of uncertain time-delay system (1 )-(3) can be converted into scaled H r= sampled-data control problem for linear system which does not involve any of parametcif'. llncf!rt.aiut.y ami nnknown time-delays. To t.hi~ end. associated wit,h system (1)-(3), we introduce the following auxili.vy sys tem:
(Efl: x c(t) = A.xAt) + x,(O) =
z,( t)
where
Xc
rB t
ft e
1'·
PROOF. The cont.rolla", n , = Fy, (ih) for system (33)(35) is given by
XcO
8,(;11 )
{(ih) = . , (t)
A, B, C, D, E and L are !.l,e same as in ( 1)-(3), W is as in (7), and 8(v, A) LS
(.onr.folled outPUf., the mat.rice>
t E [O,T]
((I) =
with Aa. Eo, H , Ho as given in (1) and (5), v and A b€,jng positive scaling para.meters s11ch that. ,\2 Eo < I. The linear filt.er we consider for (1 )-(3) is of the form:
Er
= A , I1,(t), top i/o,
(1, (0)
=
°
(37)
" ,(i/O ) = .4. d l1,(ih- ) + Bdy(ih) , ih E (O,T) (38 ) .,(1) = L , I1, (t) , lE [O,T]
(41)
=
[B, ,),8,] III,(t) t ¥- ih, {to) = [x,(of W Ad{(ih - } + 8dl ,,(ih), ih E (0, T)
(44)
[g:] ((I).
(45)
t EO, T]
(43)
where
[I-;;H
(F) : ,;-,(t)
(40) (42)
defined as follows:
_
°
ih E (O,T)
~(t)=.4:{(t)+
E )RQ+m+q
and Vc E Rq are the disturbance inputs which belong to L,[O , T] and f,( O, T ), respect.ively, y, E 'J1' is the measured output, ~I > 0 is t.he desired rlist.nrbance at.teOllation level for system of (1)-(3), .,(t) E !Rn +p+ ~ is the
B(V,A ) =
=
Now, the dosed-loop system of (33}-(35) alld (40)-(42) is given by
ih E (0,T)(35) tV e
8,(0)
= AdO,(ih- ) + E'dy(ih),
= L , O,(t),
z,(t)
(34)
is an unknown ini-
E 1RP is the control inpllt,
= A ,O,(t), t ¥- ih,
Ii,( t) (33)
= Cx,(ih) + Dv,(ih),
E !R11 is the st.ate, and
(i) )., E,!: Eo < I; (ii) With contTol law .. ,(t) = -'Ye f OT (.93)-(95) the resulting clo." d loop .• ystem, (E"I , .• ati"fie .• J (E" , R , T) <
7B(v, AlIIIIc(t}
¥- ih
= [ ~ ]",(1) + [ ~I] ",(I), t E [O,T] y, (i")
tial statf'!,
XcQ,
ing matrix R
[x~(t) 8~(tW,
Ba=[B~D],Cl=[L
~p
-L, ],
+B,III(I), '1(ih) = A d '1(i/O-)
= C''I(t} ,
eft)
lA .• + if. FoE. ]a(I), t , 7) t ¥- ih, "~to} = [",T(O) O]T (49)
+ BdV(ilo) ,
iloE(O,T)
t E [0, T]
(50)
(51)
where
H
3085
C2 =[~~~l(48)
Jj(t) = [_4 + H FE]I)(t ) +
11;(tW, ';('1,t, T) = "'( 'I. t, T) (52)
= [~: ] , H. = [ ±:~" E = [v 0J, Ea = AE"
Theorem 5 Consider the .,y.,tem (1)-(.?) .• ati.•fying (5)(6) and (7) and let (F) denotes a lineaT timc-var,ying jilr.t'- r wi.th zero initifll condition of the form (87)-(.99) .
(46 )
model:
'1ft) = [xT(t)
unc.p.rtain sy"t.p.m (1 )-(3) on finite horizon.
1J
where B(l', A) is as in (36). ::\P..xt, the filtering error, e = z· - Zi' a.ssociated with system (1)-(3) and fiUer (37)-(39) is given by the following star..e
(37}-(39). 0111' first result dealR wit.h the robust. H'XJ filtering of
[~
_4d = [B:C L], B, = [~] , B, = [B(~').)l (47)
(39)
where the dimension of filter , ne. and the matric.cs Ad. A c, B d and L , are to he chosen. :-.fot.e that (37)-(39) can be interpreted a.~ linear discrete time varying filt.er (38) followeO by int.erpolation funr:t.ion (37). which generates t.h~ sigual ~t.iIIlates between the sampling instant.s. Digital fi1t,~l's will! a 'l.t"ro-order hold are special cases of
A=
],
A"
= [~o 1(53) (54)
Note that. for any'! E
~Rn+T\~
be converted to related Hoo sampled-data control problems without paramet.er 1lncert.ainties and unknown t.imedelays, which can be solved in terms of differential Ric· cati equations with finite discr,~te jumps.
1I<'>('1,t,O)1I = 11<>('1,t,O)11 = IIW'1(t)11 where IV = [W OJ, Now, using (46)-(48) and (52)-(54), it is easy to sce that
Referenc:es
BJf; = f[f[T + .4:. (I - E~E") - I .4:~ + f["f[~ (55) cic,=£TE+WTW (56)
Bernstein, D. S, and Haddad, W, M, 1989. St.eady-state Kalman filtering with an H(
Finally, by considering the systems (43)-(45) and (49)(51), the identities (55)-(56) and condition (ii) of the theorem statement l t.he desired result. follows immediately from Theorem .'3. The next theorem deals with the problem of robust Hoo filt.ering of ntlc.ertain system (1)-(3) on infinite horizon. Theorem 6 Con"id" the ,'y"tem (1)-(3) .. ati,'fying (5)(6) and (7) and let (.r) denote" a linear time-varying filter with zero initial condition of the form (S7)-(39)' Then, given a .~calar . . ,. > 0 and an initial ,~tate weighting matri.7: R = RT > 0, the mbwJt Hoc. filtering problem a,~!wciated with (1)- (.'1) -i.~ solvable ?Jia the linear filter :F over the horizon [0,00) if there exi!~t positive scaling parameter,~ v
and ,\ fI'Il.ch that
(i) ),2E;;E. < I; (ii) With the control law ",(t) = .ry, for (3,1)-(35) the re,mltmg do,~ed loop fty.r;tem, ('£.d), i... exponentially I;;tahIe and ,~ati.~fie.~ J(Lcl,R,oo) < ''t. PROOF.Along t.he same line
a.-<;
in the proof of Theorem
5, it. can be established by applying Theorem 4 to the
closed-loop system of (33)-(35) with (.r), and t.he do"dloop system of (1)-(3) wit.h (.r),
Remark 7 It is shown hy Theor€m.r; 5 and 6 that the problem of rohu8t Hoo sampleAt-data filtering for uncerta'in <~y8terns of the Innn (1)-(3) can be cnn?Jerted intn scaled Hoo output feedback control problem with ,'1ampled mea.'1u.rem.eT/,t.~ which in1Jo/1Je.r; neither pI!mmeter ?Lncertaintie.r; nor unknown time-delaYfJ. Therefore, existing results on Hoo 8ampled-data control ,'Ouch as tho,<;e in Kabamba and Hflm (199,9) and Sun et aL (1993) can he 1Uu~d to de,r;ign robust Hex> jilter,~ for uncertain ,r;ampleddata !ly,<;tem.<; of the form. (1)- (.9). VVhen there i.~ no timedelay" in (1)-(.1), i,e, a(x,t,T) '" 0, the re"ult" of Theorems 5 and 6 will ,'educe to tho,," of Shi et. al, (1993).
4, CONCLUSION The design of robust filters for a class of continuous-time syst.ems with sampled measurements which are subject to norm-bounoeo paramet.er uncert.aint.ies anollnknown time~delays has been considpxed in this paper. The results obtained have shown that. the above problems can 3086