Analysis and synthesis of optimal and robustly optimal control systems under bounded disturbances

14th World Congress ofIFAC

ANALYSIS AND SYNTHESIS OF OPTIMAL AND ROBUSTLY OPT...

F-2d-02-5

Copyright © 1999 IFAC 14th Triennial World Congress~ Beijing, P.R. China

AN ALYSIS AND SYNTHESIS OF OPTIMAL AND ROBUSTLY OPTIMAL CONTROL SYSTEMS UNDER BOUNDED DISTURBANCES Vsevolod IV1. Kuntsevich * Alexei V4 Kuntsevich **1 1

* /,pare Research Institute; 40 Pro.'ipekt Akade.,nika GZnshkova, 252022 Kiev, [Jkru?:ne~ E-rnail: vmkun@d305. icyb ~ kiev. ua ** Iust£tutc fa'" J.~1athB1natir.sJ UniveT.~~dty of GTaz, He1:T1:ch8tr.;S6, A-80l0 Graz, A 1J,Str£a, E-1nail: alex@bedvgm. kfunigraz ~ ac . at

Abst.ract.: Analysis of dynarnic (:loscd-loop control) systcrIls nnder bounded disturbancc~ by calculating HLinirnal invariant sets is di~Cllt1Sed. It i::; suggested to HH8 th~ radiuH of a. 11lillilUal iIlvariant ~et a~ et particular llUIIlCrical nH~RSllr~ of the stability degree of a systenl a.nd ::tpply sHeh a criterion for optilnallincar control synthcsit; by rneans of lninirIli~ing thifi radius. It has been proven that. the radius of a. IIlinitnal invariant ~et rea(:he::-; il.~ lIlininllun at. a. nilpotp.llt paralllctcr rnatrix of a systelu. The sollltion to the optirna,l control synthesis problcrn ean be fOUIld~ therefore, by calculating" the linear feedback pal'arnctcrs providing the nilpotcnt ruatrix of the closed-loop t'iy~tcln. In the caHe of RcaJar controls: t.he problelll has got an analytical solution. (~opyright(g 1999 IFAC I(ey,vords: control synthesi~~ optirnal control~ invariaut, sets, bounded distllrbauce.s~ di::;cl'ctc HyBtclIlS, set val1les.

1.

I~TnODlJCTION

Set-valued rnodels ha.ve been

'\vid(~ly

applied to ~ysteIIl identificat.ion and observation, rolHlst. a.lld adaptive control synthcsi~. Set-lnernberHhip ident.ification for paralnet.er~ and perturbation~, rospectively \VOll:lt-eas<'. anal,ysi~: ha~ beeorne a regular Lool in IlloderIl cont.rol theory and reqnires cornp]et.e]y diffnreut Inethods cOlnpared to average case a.naly~is. No\vadays~ the so-called hard bound approach, \vhich u~e~ only a priori g-iV(~Il hounds on dist11rhance~, nnkno~"ll systerIl pal'alneterH and rnodel errors, attracts Illore and Inure InaJhcluatiCi
1

Supported by the Special R.Bsearch CenteT" F ~OO:l and the

contrarYI considerably less effort~ have been paid

to apply the hard-bound approaeh to a.utornatic control synthesis. The reason for thi,s goes hack to t.he necessity of calculating IniniInal invariant sets. A~ S0011 as an invariant. set ha~ hc{~n dcterrnincd, onc obtains a IIle~llre uf t.he dyualnic pI'p.cision of a 8'ystenl~

which allows

UH

to estiInate the rnaxilnal

deviation of Holl.1tious froIn a lloluinal solution and can be utlcd the sarnc ""Tay a~ a lIlcan sqHar(~ deviat.ion in stochasti(: dynaInics. Actually, t.he f~xistenc(l of a hounded inva.riant set (the property of the nlthnatc boundedne~s or dissil)at,ivit:y) i~ onc of the Ino~L ~ig;nificallt gronp pl'op~rties of tha.t class of eloxed-loop control ~YHtenu·; (dYllalIlic plantH) df~tel"tnilled by noise and/or paraII1et(~rs \vithin the given bounds. Apparently, the problern of calcnlating- an invariant set for a plant under bounded disturbances has been posed for the first tirnc by Bulgakov (1946). Fllrt,her~ Y-oshi~a\Vn (1 g(j{j) and 13arbashin

J F -NBA project No. 6243, Austria

2830

Copyright 1999 IFAC

ISBN: 0 08 043248 4

ANALYSIS AND SYNTHESIS OF OPTIMAL AND ROBUSTLY OPT...

14th World Congress of IFAC

\vhere L == A -+- BC.

(J 970) have been uHing the apparatns of f.i.VaPllILOV functions for calculating stable litnit eyelet;. Tlley ~ucce(lded to obtain 'SOlllC significant result::;) parti(:1 tlarly for lloulillcar sYHtelns~ _HO\VCV(-H' ~ the.'~e reHults revealed H01UC Hcriouo disadvantages of the approach. In fact 1 the final SIlCCCS~ efo:st~IltifLll'y clepCll(h.; on t.he choice of a particular L,yapll110V function. The early results un calculating rnininlal invariRnt sets hy 11Sl~ of an (Ixact alv.;ebraic fOl"111U1<\ hrBt prov(~d ill l(llutscvich alld Pshcnitchnyi (lg96a) can be fOllUd in K llllt~cvich and P.sbcnitchll.vi

(199Ch, 1997). Here Lhe results by K 11l1tsevieh and Pshcllitehuyj (19gGa, 199tib, 1997) arc: (~xtended to'\.vards Syllthet'is of optilnul and robl1~tly opt-inIal control HystCIIlS. \\lith the aiIH to a.pply the apparatus of Ininirnal invariant sets to tll(-~ control synthesis prohlern, the radius of a lninitnal invariant t-:(~t i~ llsed lli"4 a critcrion.'rhc key theorern clnillls that tIle radius of a rnln]rnal invariant. set reachc.s its rniniInurn at the llilpotellt par::nnetcr Inatrix. Hence the pl'obl(nn uf uptiInal control ::;ynthc~ifi takes the [orlIl of calculating the feedback pfLr(=unctcrs providing the llilpotcnt Inatrix of the closed-loop cOlltrol SystCIIl. 'The solutioIl to this pl'ohlenl fol' the case of :-;(~alar conLrols i~ caleulated allalyti<:ally.

III vic\v of the fact tha.t our rnailL g-oa]

i~

to

develop cOlnputational rn(-~thodH, wc present in the follo\.ving the Inaiu (:OUC(~pts for discrete-titHe ~YStCIlt8.

2. i\NALrYSIS OF DYNJ~::VIJC Sy""STENfS UNDER BOUNDED DIS1"1URBANCES 2.1

(~1o.,lr:1J,lat1:·ng

thp nd'll,i'rual

inva.'riani

set of a

linp'fl'" di:..;t'JJ:rberl S?J,lJtp'ln

Consider the cla~H of by the equation X n ---i-l

:=:.

AXt1

lin(~ar

+ BfJn +

control S,vsteIlIR given

P'n.~

n

=:

O~ 1, ... ~

corn pac t set.

[In

1~

i

1 ~ . . . ,.In.

(G)

,ctSSlnnc a systern (1) given by rnatriceH A and KahIHul~S criterion. Hence the ~et of Inatriees C provi(lill~ t.he f\11fi1rnent of the condition (5) is not ernpt.y.

B is controllable according- to

Introd 11C(~ t he set

"'-There the ~ll1n of Hcts is a lvlinko"vski SUII!. R,Peall, ::-let. rrn iH called all invariant Het for the SyStCII1 (12), if for .."Y E 9Jt~ the condition LX + F E 9J1 is fulfilled vvith any F E ~, i.e., L9J1 -+- ~ ~ 9]1".

Defl:nition 1. A cOlnpact set is callc~d a IninhnaJ invariant set of cl, systetIl: if it is invariant and ductS not cont:i.,in an invariant proper subset. The follo\ving t'~vo thcorerlls lul.v(~ b(l{~:n proven by K untsevi(:h a.ud f>sheni tchnyi (1997). Thp-o'rfJn 1. Under the ?.Hs\l1nptioll (5) l the Het 9Jt(L,.;f) given by fOI'nulla (6) is a cOlllpact set

in JRm,. 'Theot'P,"rn $3. 9Jt(L, J) given by forrnula (0) is a rninirual invariant ~et for a SYSt.CIIl (4). If X o 19)1(L,3), theIl for any E > 0J aB the ~t.a.tcs of Hystcrn (/1) ar(~ inside the E-ll(~ighhol'hood of m1(L, 3') ~tarting at a particular discrctc-t.iIIlC lI10lIJent \vhich depends on E.

TheorelllS 1 and 2 deterruine a convex rninilllal invariant set. at-: Hoon a..s J" is a convex ~ct. Note that jf a nl8,trix A is nilpotent, t.he series (6)

finite.

Calculating rninirnal invariant Kets of linear llOl!(closed-loop contrul) sy~Lcln~

~ta.tional"Y dynfI..lnic

under bOllndcd di~tllrbanees are discllssed here by recollecting the resultB by KuntHevich and Psheuitchllyi (1996b, 1997). sy~t.cnl~

it) p.;iven by

the equation

llBCd ,

X n+l systcrn tak(l8

n~O,l, ...

==

\\~e

~ CXn.~

cl()s(~d-loop

Xn+l=:::::LXn+l
<

Assurnc the clas::-; of dynanlic

\Vhcn a lineal' control feedback is

eqnation of a forln

fJ

(1)

(2)

t.lH~

IA,: (L) I ~

i~

\vherc X n and [In cU'C a st.ate J1J,-(hlnel1:::1ional vector ~ r(~.sp. a t:ontrol p-dirnellsional vector, at the nth discrete titne iur;tant, A and B are the lIlatricCR of the rc.spcctive dilucn::..;ion~ and F n is an H1.-(liIuCllSional v(~et()r of l)(nln(l(~d (listllrhant:(~s:

J is a convex

Independent frotH other rnatrix properties v-rhich IIlight be needed later, L is &~sluned to be a SchurC~ohll lllatrix, i.c' l all eigcnvalnes are in the open unit circle:

,

th(~

=::::

A''2.Xn

+ BUn + j-!"'n.,

11,

==:.=

OJ 1~ ... ,

"vhere all the notations arc the H9.l1l(' as in (1), except An is a Inatrix of tirne-'varyiull," pala.nlC1.(~r~: o

(4)

An

=:=:

A

+ bAn~

n == O. 1, ....

2831

Copyright 1999 IFAC

ISBN: 0 08 043248 4

ANALYSIS AND SYNTHESIS OF OPTIMAL AND ROBUSTLY OPT...

14th World Congress of IFAC

2.4 In1)Q,T1:ant set..,' of specia.l tllpe 'fI,on!£ncu','

o

I-icrc A i::-; a constant rnatrix and

bAn

E {;21 =

{8A:

IlfJA~I:;

rr},

n.

=:

O~

,'i1f8t~'fTr.8

Consider t.he class of systelIlt-i given by the equa-

1, ....

tion .A.. HSllnH~ linear fc(~d hack (3) fol' this cl ass of s,ystenlH and cOllt'idcr the clo~('d-loop cOlltrol S:y~tcln:

(7)

(11) vlhcrG ~ (-) ic-; a nonlinear vector function 1ft"n R m . sati,sfyiug the inequality

\vhcrc

(12) o

Ln

=::::

A

+ cn + tAn?

n

=.

and the disturbances F n are hounded hy (2).

D, 1, ....

o

Agaiu) aSSllIllC L --= A + CB is 8. Schul'-Cohn IIlatrix (S{X~ (5)). Introduce th(~ €-ll(~ighborhoodof t.he set ~:

~T here <:3

iH a 'lui t solid Hphcl"c with the center at

tJnder the condition (12) ~ a SystCIIl (11) has a. invaria.llt set. Ho\vever, onc canllot calculate thi~ ~et explicitly in contrar:v to the ca.-,-;e of linear S.ystCIllS (4). OIl the othel" hand, ~iucc one has got an upper bouud for the vallleH of ~(Xn), it. bccoInes pos~iblc to estirnate the radius of a lninirna.l invariant set, bOllnd(~d

the origin, and the Het

9Jt(L, J=)

---1-

::=

r(9J1)

9J1(L,~) +.=.6

+ EL6 + EL 2 6 + ... " (8)

K ll11tscvich and PSh(~Ilitehnyi (1997) proved that (8) is a IlIillirnal iIlvariaut set for the class of llOlltltationary :-;yst.(~!ns (7) for any ill(~q llali ty

1- q

iT

E Hat;i~:,;(ving

tIle

(1'

:

X E m1},

by the fono'\vinp; iterativc' procedure. ASSUlue .9J(0 ==.J"~

r( 9J1 o) == r( lr) == p-

+

Calculate 9R 1 ==:
~.

The

l'rLdill~

of 9Jt 1 is

obvjou~l.y

r-(9Jt 1) S; r( (~))

+ P :S () +

fLp.

f:

< ------

= 11lax{!lX I~

d(J)

~AJtc;r

+ E'

where d('J) i~ the dicunctcr of the set J' and (x is a C()n~tallt providing the flllfilrncuL of the ineqnality L n ::;; ()'qH. rrhe latter is correct. for Cl lnatrix satisf};iug the eOlldition (G) and a constant er > 1.

2.3 Inva.,·ia'll.t sets of d,!)na1n'£r;

8JJ.··dern.., 1j}ith

c~tilnating

the ra.dius Y"(ffit k ) by its up2:,i=O, ... lk 1".1, estiInating r(9J1,l,~+l) is stl'aightforwar
(J

'111€O'reu,. 8. The radius of the lninilual inva.riaut set flJt(
bonndAd

'f},oulin t-:nT pa'f't,.c;

The follo~"'ing class of SyHtcIIlS have been analizcd by KUlltscvich and Pshenitchnyi (199ua).

\vhere L rneet.s the couditioll (5) Ctud cfJ (-) is Cl. ------) R'Tn, there\vith

continuous vector fUllctiuIl R 1n

(10) i~ a cUIIlpaet

set ill

ll{'m.

Here e1(-)

IneallS

closillp.;

a. set.

It is ~trai~htfor"val'd that the syHteYflS (9,10) and (4,2) :it c
l\ nal:'lsis of 1l0Uallt.OllUlIlOtlS ~:V~tclIl::; v.,rith bOlllldccl lLonlillear parts leads to the sirnilar rc:snIt (S(:c K \l.llt.~evich and Pshenitchnyi (1 ggoa)").

2.5 M'ini'fnunJ, l'adin:; in1JaTiant set lfercinaft.er, t.he radius of a tninilual invariant set (or an cstirnatc for thit; .set) iH Hoed &'-) a c.ontrol pcrforlnaIlcc criterion.

A rninilnal invariant set, ~'"hich has rniuilnal ra.dius arnong- all other invariant sets, \vill be called a rnr:rl:irnu.fn 1'adiu.r,· iuvar-£anl ~,,·et. C~onsidcr

a control systenl (4,2) vlith 11

L==

rrIcCining

L

takc~

~l

1

II~ ~~

() 11

.(~l

tlle Htandard

(13)

Frobcniu~ forlll.

Think of L as the Hlun

L

=

Lo+ L

n?

2832

Copyright 1999 IFAC

ISBN: 0 08 043248 4

ANALYSIS AND SYNTHESIS OF OPTIMAL AND ROBUSTLY OPT...

of a uilpotent C-"vith the nilputeucy index ulat.rix

I~~)

14th World Congress of IFAC

TheorCTll 4 clairns the luinirnlllU raelius invariant set at L ::::::; L0 7 i.e.)

Hi.')

L m.

1

-=.:.::

Am,

I

bm,C ::::::

o.

Hence

L o == 110 () 10 0 1

'

and a luatrix

For vector controls: the synthesiH problcrIl canIlot be solved that elegantly and easily it is a particular ca~c of a l110dal control K:yllthesis p1'()hl~ln (se~ Klluts(-~vich aud I(untsevich (1994 1 1990) for instance). The solubon of the s.ynthesis problcIll in this ease ccrtaillly eaullot be given ill explit:it forIn and \viIl require solving rather (~OTnpJcx ntinirnax

l}/,r:orprn 4. DJt( L o , J") i~ a rninirn111H radius invariant. ~et uf ~ystcrn (4~2~ l:j).

probleJlls. rr"'heorerIl 4 claiInti that InininnuH of the radius is rCRched at the Ililput.ent par<11Ilct.cr IIlatrix L =

;).2

L o·

SJJrdhF:8i", of

cont-rol . . 'y,<.,'tcrn,"· u;ith

optirnal

11.ucertai·fl pa.'ra1netc'('8

PROOF. Considcr the sUIn of the fin·.;t two in fOTlnn]a (ti): ~1Jl(L;~)l

=?::

~~

+ L~~

=

(~+

LoJ)

Re1na.,.k. Synthe~iZtinp; a contrul 11\1 hich provides a lnini lInlIn radi us invariant Het foT' a closedloop sy~t.cnl~ onc: obviously ha.."') to check fir:::-;t

itcln~

l

+ L'm,~'

the robust stabilizability of the given

clR~~

uf

Ohviollsly, Lm.~ == {O}) if \I F E J Li=l ,. _. ,m 1,11 (L rrh(~ latter is trll(\ iff (11: ::=::: 0, 1: == 1, ... ,'ftL TIH~ ~Hun uf IlOIlCIIlpty sets is larg-cr thau each of thcrn, i.e. ~ 9J1(L o , J") 1 c 9)1(L, 3")1. Thcreforc~ r (ml(L o, 3')1) < r (9)7 ( L, J")t).

Unfort.ullately, the robust stahilizabilit:,'{ problcIll ha.Ll not been solved yet. "£hat i~ \vhy the robust stabiliz;al)ility of the clas:::-l uf systeIU:::-l ,ulder consideration is as~ulned here a priori.

SiJnilar eOllsiderations for the consecutive sets 9)1(L, tf)Ii~, the ~urn of k + 1 itcrns in fOl'nntla (6), lead t.o the salne result, Uy indnction, OIlC obtaill~

fiyst{~InS

The

H~e

of

ThcOT(~rIl

4 will be

~ho\vu

s.YHten)~.

IIcre~

the consideration i~ extended to the class of \vith Ullcertain pararHcters asslllning t.he

cor-:.flicir:nl...,' of L r!'l,ay take thei '(" 1Jalue,." in a .qiven cornpart

~on1Je.rr

...e.

Definition 2. AHs11lning S(:hUI'-Cohll cOllditioll (5) is fulfilled for every rnatrix L E )2;, a rnilliYllal invariant set. of the class of systerIlS ('1) with 1111certain para.lnctcr~ is the union of the luillilnal invariant 8cts (:alcllla.tcd for every particula.r Inatrix L froTIJ the set £:

ill the Ilcxt

scctiOll.

J. SYNT'I-IESlS OF CON'TllOL SYSTElvIS UNDE,R B()tJNDED DIST'lJll13ltNCES ;3-. -I Oplirnal cont'l'ol 8l/nthp-s'7.s 1vith.

."·et

m1(£,~) ==

U 9J1(L~ ~)LE£.

A-;no10n paTa111ef,el'

In particlllar, if

'f'J,att·i~T.

By virtne of

.£ =

Thcol"(~lIl 4~

the synthesis problclll i~ r(~dllCcd Lo (:(tlculaLing a feedback Inatrix C opt providing tlH~ needed property of n· Inatrix L. Such lua.trix C opt. alvlays 0xistti, if systerIl (1) iH cUIlL]'olhthl(~ accordiIl~ to Kahnan's criterion. Cah:lllating: CL fee( Ihc:u:k IllR,T,rix is t h(~ ea.:..;iest for scalar coutrols. In this ca::,K~~ it is a.ppropriate: t.o cOllsi(lcl' luatl'iccs A and B of the standard f01"111

A=II:}__ :__ ~II, ~I A~,

11=

JI

v;,;rithout loss of Rell(~rality. Obviollsly, IIlatrix L ==-A + .BC is of t.he ~taJldanl [orrn (l:j).

<_COllV

1

,,--1,<,. 11\

{L1:}~

~rith 1\l verticcs~ calcula.t.ing a Iuillilual invariant set bCCOII1CS le~s difficult. The foHo\\.Ting re:--;nlt wa.s first prp.se:uted hy KUllt-

lueaniug £ is a polytape

sevieh (199U). TheOTe'ffl, 5. If the condition

every L E

)2;~

9Jl(£,~)

([)) is fulfilled for

then :=;

(,()llv{~1)l(Li,:J): i

==

1 r ••• , N}

is i) nn invaria.nt cornpact convex Bet for tlYStern (4) \~l.ith uncertain paralnctcr lllntrix L E ,.e. and ii) a Hon-reducible upper hound convex (-~sti­ rnate for the Inininlal invariant :-let of the cla.~~ of Sy~tClllS.

2833

Copyright 1999 IFAC

ISBN: 0 08 043248 4

ANALYSIS AND SYNTHESIS OF OPTIMAL AND ROBUSTLY OPT...

Tll(~

14th World Congress ofIFAC

rnost siguifieaIlt curollary fronl rhe above i::l

TheOTe'lfl 6. Under the conditions of Theol'ClIl 4 and for the given set-valued parR.rnct,cI' estin1ates (16) thn feedhack veetor

ohvious:

l

[;o'FollaT?J 1, UIHler the cOJ1dition~ of TheorcIll

5,

o

the raclius of a COllV
(Jopt = -L1 m

is

l'hc ra.diu~ r(ffi1(£.,~)) call be calculated directly hy (~va.l11atiIlg the I1Urln~ uf the vertice::-;. Howcvel"~ the l1~e of sHch a JJlinituiza.t.ion criterion i~ lirnitcd t.o low dilne!lsion~ Tn,? sillce one has to pay t.rcnl(~n­ dOllS CUlIlputatioual efforts for

hi~hcr

. (:OH1,7

'l=l, ... )V

for the pl'obleTn (15).

PROOF. Further in the prouf, the index

-rn. \vill

be ornittcd. Introduce the notat.ion o

'P(~A,

dirnensiollH.

8b, (;) ===

IIA + 6.i! +

0

(l) + 8b )(;JI

(17)

for the objective function of t!L<' rniniInax: prohlelLl (lG). Here oA E 82l and 6') E c5b. A~S\llne

Alt,(~rnativcly, eOllHidcr t.hc; class of Hy~tcTnH v..!it,h scaJar (~ontroJ aud uncert.ain paralIlctcrs ..l4 m E 21 m . alHl lJ.m . E E7 1n , where Q1m.

a, lniIlilniz;(~r

0

/h rn

o

0

C opt = -..l4/b and substitute C hy

CO'Pt

in (17) to

obtain

{A~n}

(18)

is et polytope and b·rn. =

{b'nl. :

f!.m. ::;

IJ.,n ~ h1n } ,

Eqllival(,Htly~

is an interval, j-1~n.) 'i == 1 ,.IV, ar(~ the verticc;-.; of a p:;ivcn polyt.opc 2( 'rn and f!..m ~ 6111. arc g-i ven cOllstants. i

•

•

•

Let. llH point tu the f(·~a.Lnr{;s of Ho nilpotcnt lIHl.trix except thot;e ruakillf]; ll~e of eigcn va.lues. Consider the -(n.-tit degree of a lnatl'ix L of the standard

(1::1)

fOJ"tIl

11 L'[;, Ill' Lm=IIG(L~),,'

{i(O) =0.

(14)

It is straighLforw
(IG)

lllin UHi.X {IIA m . -t- lJ n1,(711}· A., .. EQi""

C

0- 1 0

introducing; the notation tiL == bA - 8bb A~ \vhere ~L take~ its valuc~ on the set 8£ ~ cS21 \vh(~n 0-

oob

1

0

.A..

Set 8£. iH a convex COl.l:lpact set, since tbe set~ 821 and hb aI'~ convex and cOJupact. FurtherIIlore, these two sets contain the origin , hence o£; contains l.1Jc urigin tuo, This iIllpOl'tant property ,viII be llsed l
cp(8A, 8b, C opt ) Le.,

G"opt i~

a rninirnizer for the pl'oblcHI (15).

Substitlltc C hy

b,,, Eb."j

It is ",vort.h to note that the optiInizatioll problcll1 (15) appcar~ to he quite the salIle as cOlu:~idcred ill I(lllltsevich a,nd I(uutsevich (]994), where another perfornlallC(~ (Titerioll wa~ ll~ed.

Expluring- the \vay the HYllthc~iH ploblclll has be(~ll solved a.bovc analyticfLlJy for a syst(~ln v;,rith knowll o

paralucter Inatrix) calculate the center . .4,nI. of the ~et 2l m . as the C(~llter of r,hc; tniuirnal upper hOl1 nd solid sphere con tainiug the polytope a.nd the C(~Jltcr b'l'l1. :;:= ()'[)(~m, + 6m J of the iIlterval set b (here, \ve aSSHIIlC () t/:. b). Ko\v) repres(~nt the given pa.rarnetcr eHtinHtte~ iu thQ centralized forlll 2lu~

o

0

::= ...4n~

+-

82l rn n.ud bm, ===

llm

+ ~bm;

(1(j)

Q

\vh(~r(~ b211'Tt "rlH~

==

follu\-ving;

pro1 )}Clll.

conv{ . A"~1 - A·rH theOl'CHJ.

:

1.

=

1~ ... ,.Z''l}.

gives tlH~ solntioll for the

~ cp(bA, 8/), (;),

(;upt

+

(71 and ol)tain 0- 1 0


=

/IbA - 8bb

. .4

0

I

(b

+ 8b)C1 1f.

Next, take into account the llotation.~ ftcceptcd il.l (19) and rewrite the ahove in the for-In:

No~r,

recall t.hat 1) is strietly llC)lll.tive 01' st.rictly positive on b (0 tf. b) and 8£ cont.ains the origin. lIence for any Cl, onc can find a v(~ctor /j L* E /j£, such that

TheurCUl G gX~!lcralizes the corre~pondiIlJ,!; result pHbliHh{'d by KUllt~cvich
Obviously) the n.Jhll.stly opt.ilnal control synthesis problclll should have ueen cOll~idcred at a pararIlctcr lnatrix hclong-ing- to et Sehllr-Cohn tict-.

2834

Copyright 1999 IFAC

ISBN: 0 08 043248 4

ANALYSIS AND SYNTHESIS OF OPTIMAL AND ROBUSTLY OPT...

14th World Congress ofIFAC

H(),,"vev(:~r,

it scenlS ahnost iUlpOSKiblc to overcorne the rnajor difficulty- of applyinr; thif) (:oHstraint, because there is no c:oll~trllctive (in viev...r of COIIlplltaLioIls) dcseript.ioIl of a :-\ct of ~tablc Inatriccs. IInl1ce, (In(~ ha.s to check the robust stahility of the sYIlthcH.ized cOlltrol systenJ aJter calculating: C:ojJta If the rohnst stahility conditioIl~ arc nut fulfillod, the only ""ay is to run a sct-rnernbership (~stiIIlai,illg procedure anCVl \vith the aiUl to l'CdllCe an uncertainty in parCl,rnetel's and then recalculate

V.1\'1. 1-( uu tsevich and .A_. V. K nu tsev ich. Robust, Htahility aua,l:rsis and control synthesis nndel" uncertainty. Syste'rn.." Sr.ience, 22(3):17-28, 1990. \'.~'!. I(Ullt~cvich CLud B.N. PHhcniLehllyi. Invariant. a.lld ~tn.J.ional'Y ~(~ts of nonlillcar discrete Sy8t.CU1H under bounded disturbances. P,,.ob~ !t-;rflY '(,ll)'((lvlcniya i info'{''fnatiki (Jou:rnal of A '(1to,.,nnJ1~on

(7apt .

4.

C~ONCLCSION

and Tn!01'1nat.io'fl,

~S1rjenr.f'j'l)~

(-1-2) :35

4G, 199Gb. ill Itnssian. V.1vL Kllut.::-levich and B,N. PShCIlitchllyi. :!'v1iniIual invariant sets of dynarnic ~ystc:nls with bounded dit-;tllrba.nce~. CybeTnct1:cs and ~')?J.'jte'ln,,, Analy8'i..'i\ ;12(1 ):58 (j4~ 1990a. TraIl~lated fruIn KibeTnetl~ka. i 8isternnyi Anali..z No. l~ pp.74-81: 1996, V.lvI. !(llntscvich and B.~. Pshcnit.chnyi. :Ylinilual invariant. sets of discrete systelIlS ~Tith Htablc linear p:1.-rts, P1"OblernJ) npTa.vlenjya i i'njo'l"rnat1~ki (.Jo1i.171,al of A1J.tornat£on and Info',"rnat£on /:;cience...-j, (1) ;81 -91, 19D7. ill Rnssiau. 1\11. ~/Iilane:-;e, J. NortoIl I H. Pi(~t-llahaIlicr, and E. "\i\laltel', editors. _Ro11..nd'£ng a.pP'f'oar;h,Po,'; (,0 b'rlstpJn ident(fir.at1:on. Plcnurn Press, Nfn,v- YOI·k Rnd London, 1996. E. vValt(~r and L. PrOllzato. Idt:ntift:cation of Pararoet'f'ir: AIodelb' frorn. E,7:perirnental Data. 1

By acccptin!!: the hard-boHud COllcept, one llccesHarily ha.s to refol'ln1l1at(~ anc~'" all Tnajor anal.)'THis and control synthesis pro ble1I1H and al~o onc: faces IH~'V unique problctns having- llothing; ill cornrllon \vit.h tbose rnet ",,.hell using- Ktochastie Ill0dclH. The purpose of this pap Cl' is finding; an (;asjly calculated r!lCaStlre of a systclYl reaction to bounded disturbance...". The pn~s(-~nted Inctbod~ of HllalYNis and synthesis ruake llSP of tJl(~ pCl'fOrIllancc eritcrioll, which is Iuinirui;c;illg the l'adiuH of CL Ininiulal i!lval'iant set of a. dynanlic (closed-loop control) :-;ystCIll, r-fhcse Inethods arc applied to linear and SOlnc typ0H of nOlllincar systcrns. [fhe cla.'-'b of l10nlinear rnodels llIHlcr (:oIlsideratioIl is l"CBtricted to tho~e \vith strictly bounded llonEnear parts. It i~ v(~ry iIIlporta,nt to extend this class.

T.

Springer-Verlag, Berlin Hcidclbcrg~ 1997. )ro~hiz;awa. 8tab1:Uf.1J lheo'r"y b:rJ Liapu.'nov,1.-; second ',nethod, NUlllbcr 9 in Publ. of the l\:lath. SOf~. uf Japan. The lvIathclnatical Society of Japan, 196u.

A lllore (lifficult (-llHl COIIlIllonly faced pl'oblclIl of the robust stabili~a,bility relnains yet. HIlsolvccL How·~cvcr, the existillg control syuthosis rnethods proved to he applicable (and ::-;oluctilIlCS very cffi~ CiCllt.) to a "vide class of feed life control ::-;ysteIIl~, where the hard-hound concept is the Duly acceptahle.

RcfcrCll(:CS

E,.A.. Barbashiu. Introdurtion to the theoTJj of ,4}tability. v\tolten-.-Noordhoff, Grouingeu 1970. U. \l. Dnlgakov. On the accllrnulatioll of dist.urbances in lillear oscillatory systculS 1~"ith (:011Ht.a.nt. paraJ·n(~tcrs. Dokladu Acad. },'r.i. URSS, 51::143 '345, ID4ti. (in RtlSSlan). A.. V. K tlutsevich. Sct-lllClnber~hip identification for l"OI)lU..;t (:ontroJ. III (/ESA :96 IJl1AC/5 11;1ulh:confwrenr.e (L-il.lt".-Franre; 1 996), v()lllnH~ 2, pa.u;CH 11G8 1172, Lillc, 199G. A....'l. Knutsevich and V.l\1. KlLllt~cvich. Lill(~a.r adrtptive control for' llonstat.ionary sy~tenJs Ulldel' bouuded noise. /,?/8fpJn and Cont'f"ol Le.ttPT.,,:, l

7

:11 ::)~) 40> 1997. V.M. T
2835

Copyright 1999 IFAC

ISBN: 0 08 043248 4