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On Right Invertibility of Nonlinear Recursive Systems
2b-14 2
Copyright © 1996 [FAC 11th Triennial World Congress, San Francisco. USA
ON RIGHT INVERTIBILITY OF NONLINEAR RECURSIVE SYSTEMS VUc Kotta'
• 11lst.itute of
(.'yb e rn e ti(:.~ .
Akadeemia t ee!!1, Tall1. nn. 0026,
Es to1~ia .
F-mail:
kotla @ioc.e.e
Abstract.. The right. invertibility pro blem is st.udied fo r a daso of rec ursiv 4~ lion linear systems (RNS), i.e . for systems, modelled by rec Llrsiv(~ nonlill t~ar input-output equat.ions invo lving o nly a finite number of input values and a finite numb er of o utput values. Th e con c.ept of delay orders and the spec ial ease o f right inverl ibility, the notio n of (d J, . , J II )-fo rwarJ time shift (FTS) right invertibility, kn own rur discretet im e nonliu ear systems in state space form : are extend ed t.o R NS. Necessary and s uffici e ll t condit.ion s fo r ln cal (d 1 , .. ' Jdp)-FTS right invert.ibility are propost ~ d. Finally, il: is s hown ho w 1.0 gt 'nf: ralize the :;pecial case of right invertibility - (d. , . . ,
Keywords. ltol1l illCiH f t";cursivt' system, in ver1.ibility. inverse sp;tem
\. INTRODU CT ION The fundam ental ques tion s of tl1 e 4 ~ xistence and the COIlst. ructi o n of righl. in\'~~ rses for uonlinear discrcte- time d ynamical system!> art' addr€:,.l:)scd ill thi s pap er. Such a
right. iuvNse is intuit.ively und nrs tood to he a. second nonlin ear sys t em Sll ch th at. whfm tlw original system is appli ed in seri es wit.h this rig ht inverse, t hen its outp ut.s are equ a l t.o tli (' inputs of t.he right iu verse system. B~
ca use of' the illherent. delay typi cally found in dynamical sys tems , in it. great. Ilumbf:"'r of cases 110 s uc.h invers io n is p ossibl t', and tht~ pwhlt~1Tl is of liJllited inteH~s t. . Greater g<'ll e rali ty is o hta in NI hy cO Jl s id ~ rifl g a notioll of forw a rd t.ifTIt' shift right illv!,~ r ~c ill which 1.114' input to the r ight inverse l:)ysi:elll is not j list t.ht' refe rl' IKt' signal Yr eJ (t ) but the fefr: r(' llce signal at some fut,ure time inst a nt t + (r, i .c. Yr l'! / (I, + n ). The determinat.i on of th e smallest possihie valu e for rr is a questi o n of prdcl-ical and th cor ~t.ical import a nce. In cas(" of Illlllt.i--outpul sys t.ems , the value of I.hl:' s mallest possihk 0 , ill ~f> lI e ra l, is not t.he sanw
di sc r~t e - tirne
systern, delay orders, right
for all components of the rt fcrence sign al. A very impor tant concept in treating sys tem inversion from f.his generaliz-ed point of viow, is 1,he COll ccpt of delay ord(~ r s.
Except t.h e pap"r (Morhac . 1990) on finding the in\:erse Vo lterra represent atio n , most pre vjo us work 0 11 this snbj ed ha.':' r.oncentrated on sys tems h
19 90; Grizzle, 1993; Kotia, 1986; Kot.ta, 1990; Monaco and Norruand- Cy rot , 1987 ; Nijrneij er, ] 989). The purpose of our paper is to s tudy the fo rward t.im~ s hift right
invertibili t y pro blem for a dass of systems described by r e:c uf~ iv(~ no nlillear input ·o ut.pu t equ a tions illvol v in g only a fillit,r' number of i IIplll valu e~ and a finite IlUmb e>f of o utput. values. Such sys te ms are called recursive n Oli lint.:a r syslems ( R:"lS) (Ha[lltncr, 1984; Leontaritis a n d Billings , 198.5) . \/lle sha ll f! xte nd Lo this class of nonl in· car sy stl ~ m s t.he conce j.>l of delay orders :md the notio n of (d J , • . ,dp ) - FTS ri ~ ht invertibility. The latt.er is a sp ecial case of F'l'S right iJ1ve rtibility in c.ase of whi ch
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th e smallest poss ible values for (1 ) ' . , Up may be rea lized. Note t.hat in (BMt in et al. , 1993) for single-input single-output RNS the qu estion of def\.d- beat control has been ~t udit~d . Th e latter prob lem is closely related t o the right inwrtibility problem . Howpver , unlike (Rastin el Ill., 199:1) W~ consider I..llt' rnull.i -input m ult.i-outpu t !;yS 1. ~ ms and CUllcentrat,€,: Ot] t.he lo('.a l solutions around an equilibrium point. of t h ~ system .
equality yO = P( y O, Un) wh"'e y o = (yO ,7' , . . _, yO ,T)"" UO = (u.0 ,T, ... , uO .T)T.
From n ow on, we co nsider RNS (1) at. nonn egative tim e steps in a. finite tim e interval 0 ~ t ~ tF under iui t ia l conditio ns
Denote by
2. DESCRIPTION OF RECURSIVE NONLINEAR SYSTEM In th is secti on , b e:-;idl~s recalling Uw notion of rec ursive nonlinear sYf:item (Eal5tin et al. , ) 993; Hammer, 1984), wc estab lis h so m l~ no tatio ns and inl,fod w:.e some pre liminary mat.l'ria.L We denot.e by 5'( Rm) t.h e sl2:'l. of fLU two-sided infinit.e sequ ences of the fOrlll
{ =(t)} = (
, =( - I ), =(0 ), .,0), z(2) , .. .)
where z(l ) E R:n for all integer:',. l. A dynamical syst.e m i, a map ~ : scum) ~ 8( 81' ) ("(i)l ~ (yet )) which tr ansforms input seCJIl ~nr;~ {u(l)} illl, u output sequence {v(l)). Given two sys!.e rn s ~, S(lI"' ) ~ 8(RP) and E,: S(81') ~ S(Hq) , denot(' by :S', 0 E, : 8( R'" ) ~ S( R.9) t.he sy ste lTI represen Le d hy t. he ('omp o!:!ite map . A finit.f' suh seq uellce of t, hf: infinik :-;equencf' {z(t)} be-twcen tinw i nsta ncc~, t a nd t - T SI,H',ked in t.h ~ co lumn vecto r is deno ted by
w.
.,,(1) '"
=
p(y . - I ( 11.- 1.) 1-}_'
I- v
(I)
w he re F Rl l l' + lIfH I--T RP is all an a lytic map and I < Jl S; :::xJ: 1 S v ::; 00, then t he syste-m 1: is said t.o have a 01l115al finite di ull:' llsion al rea lizatiun .
D efinition 2 .1 R ecursive llonlillear system . Recursive non/iu ra1' sY8 tcm ( R NS) v; It ·""Y8 tem which I,as a cflllfi ul fimL i' diwou;iona[ rea lizatwlI oJ the form (1). D efillitiulJ 2.2 Equilibriunl point . '/'he pair' of conslunt
TJu{tUS
(tl°, yO) is ca,lIed th e equ'ilibritnn powl of tht (IL l! , ',!l ) Batisfies the
a (J.l.p + vm )-dir ne nsional vector (yO,T , . the system (1) has as inputs
t he ''''tu pnce u = {u(t);O ~ I ~ tF}. Throughout the pa per we sh a ll adopt a local viewp oin t. More pf(~ c isely , wc work aro und an equilibrium p oillt (u",yo) of the syste m (1) , l et u, deno te hy U O (resp . U) the sd of r.o nlfol sequell'es " {u(t ): 0 ~ t ~ t F) (resp. Utt~;) sucb tha.~ the cllntrols u(t ) fo r every tare sufficien!.ly close to u· , i,e. that 11 u(t)-u· I I ~ 6 fur som e fi > 0, Allalogously , let us ,knote by y. (resp . Y) the se t. of 011 t PIIt. sequellr<.'S (yet ); 0 ~; I ~ I,,} (resp. )','~"' ) s uch tbat the o utput. ~ y(t) for eve])" t are sufficie ntly close \.0 yO, i.e . that 11 yll) - yG 11< , fo r sorne, > O. Finall y, let us de not.I ' XO t he ne ighbq urhot:1 of ;r;o s uch tha~ fo r every x E X ", 1I :r - x" 11< -; fo r some -y > O.
=
uy
For difference equation (1) Illld ~r initial conditions x(O) , as long as F is a wdl-dcfinf'd function of /l..,",P + VIn , t,hP.ft" is no problem rcga.rd ing f!xi stence and uni4ueness of itl' solution y(t; 0 ::; t :5 t.p), for a rbitra.ry control sequenc(' u E 11 0 anrl arbitrary initial (ond ition x(O) E XO. Such Cl solul,ion will be df' noted as yH 1 x(O) : u ) which is a sho rlhand writing for !I(t , x(O), u( O), . . _.u(t - I)).
Z:_r
If T ~ 0 , it. is unders t,ood thai, denotes a n empty subsequen ce of {~( t)).lffor e very in put sequen ce (u (t)] , th e corre~poncling output ~e4l1cncr {y(t)} of th e ~y~tcm E sa.tisfies t he equation
:1;0
yO,T I .(to ,T : ... 1 uO ,T)T . Then
3. THE DELAY ORDERS WITH RESPECT TO THE CO\'TROL For discrete-time Ilo nlinf!ar ~ys t em s J describ ed by ~t.at(' equatioJl)), t he dd ay o rders d" i 1, .. . , 1' , (referred also in the literature as the char;Ldcristic num bers, the relative orders or the dead tilIll ~S ) with resp ect to the (.o ntrol h avl' been defined , one fQr ~ acb output compon ellt.. These system stru ct ura l para meters tell us how m a ny inherent delays tl lC re a re bet wee n the ith tornpo nent. Yi of the o utput and t.lw contr'.>l , or equiva lent.ly, for how many fir:o:;t time ill stances Yi jf:) complet.e1 y defined by t hl' initial cundit.ions a.nd whi ch is t he firs t tim e instant fo r which th e potiSi bility arises t.o change Yi a rbitrarily. tll this secti on we sh a ll extend ~h e concept of delay orders to the dass of R NS . Definf~ t h t:> it h ,-omp onent of Jo' ill (I) hy F;.
=
7·t f' u rs ive llouli n(~(I1' ,'iys/ em (I) if
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At. firsL sight it. m a.y seem t.hal. ont." c:w defin~ di as t.he !o;mallest. p ositive integer k such (.li at ')
(
i)t((l - k)
F(y' - I /1' -1 ) , I - I" I-v
tt,,,
is not identically Zf'ro. Using t.he abo\'l~ definitio n, a RNS (1) wit.h del ay orders d l , ... , d p admits for i l, ... , p a representa t,ioll o f I.he torm
=
y,(I
+ d,) = F; (Yc':t ~; , !t (t),
[I,'+1._v) '
(2)
However , th ~ above defi.nit.ion is noL , in general , in co mplete correspondence with the staLe spaw formulat ion of tJlf' concept. sinct~ Lt. dn e~ not ddhw rUI' how many firsL rirJ'J E' insLann;s Yi is cornpleLcl y defin ed by the initial condit,ioJls ohserve that (2) co ntain..., y (l +d i - I ), ... , yet) whi ch are not the part of illitial C{ .nditions nnd may depend 011 co ntrol. Th l~ full owin g f'xalllplc serv ~ s as an illus trat.ion.
Example 3.1 Crm.'lfder th e ·<;Y5I r m
= "l(t y,(I) = ",(I. -
1/,(t)
I)
+ ",et -
2) -
=
Yi (l
U3(t --
yet + 2) = F/(y(t), Y'-~;+I' u(t), U,':;+I)
= Fl(Y.'~:, l ',':,~, u(I)),
I)
If this vec lor .i~ non zero on all open and dense subset. of Y x 0 , we set di 2; oth' ~rwise we continue with
2)y" (i - 2).
=
l(t Yz
H ow ~~ ver,
actu a.lIy there possihility to c.hauge y'.!(t I- 2) a.rb it rarily sinc;e (4 )
+ 2)
= V,' (yI '- -I U'-I) IJ' t-v'
+ 2) =
(Jj
(6)
In this way Lhe numb~r di - If it e xist. ~ - determines the inherent delay bd.ween the i1lputs and th ~ it.h output.. A R NS ( I) with delay ordel's d" i represent.ation of the form
=
I, ." , P admit. a
and if we replace Y3(t. + I ) in (j) by I,he RHS of (4) ill (:3) we see that Y2(t + 2) depends ('.olllp letely 011 t he initia l c.ond itions· Y2(l
(5)
2)!J2(t ·- I)
By the abovf> definiti on, d 2 = L 11 0
,,'.-'1/ ) • = Fi ' (Y'-' t-Il ' v,
Apply aga ill a. forw ard shift operat.or to equation (5) a lld replace in the lat ter yet) via the right hand ,ide of (1):
Co mpute
is
+ I).
COfnput.t' in an a ll a logo us fa..... hion the derivative
",(I - :l)YI (I .- 2) + Y3(1 -
y,,(t) = - ,.. , (I - I) -
From the analytirity of the >y.tem (I) it follows that either vector dF/ Of du( I) is non zero for all (Y.'~~ , U:~:) belonging to an open alld dense subset OJ of Y X 0 or this v(~cto r vanishef:> fol' all (Y/_-t41 , Utt~~) E 1;' X U. In the fir::d. ra-se W f" define di J whereas in the latter case we cOHt.inu(' by observing tlwt the fun ctio n J/~t does uot. depend o n u(t) , i.e depends completely on the in it.ial conditio.tls and so we may writ ~
(7)
-1(,(t - I)YI (I) - ",(t - l)y, (t - I ), o r ill thE' vec tor form
N ~x t
wc shall give the propN ddlllit.ioIl of t he delay ord e r~ for RNS. The forward ti11l e shift operato r 6 is denll ('(1 as 6y(l) = y(l + I) . Apply t he o llc-slep forward ~ "ift. operal,or 1.0 etlllatio n (I) a.nd f._~ place in the latter y(t) .... ia the initial conditions, i.4 '. via t he right hand side of (1) in orciN to obtain
y( I
+
YI(t +, [
y,(1
dJl]
= .4.(x(t), u(t)).
(H)
+ dp )
where
I) = F(y(t) , 1'','_-':+1 ' ,,( I), [!,'~ ~+ I)
= f 'I(Yt-' - !"'
(.'-I
'I-v''//.
Denote 1.lU' ith componem of /0'1 hy i = 1, .. .. p the d~rivati vt'
It )) .
P/
By definition of (Dastin et al., 1993) a prediction m odel and compute for
with predi ction ho ri zon d for a R NS is a rc("ursive map which a llo ws t.he computati on of thp. output at futur e
2408
tim e inst.ants t + d from inputs up to I.jJn~ t and outputs IlP to time t - 1. According 1.0 the a.bove definition the represenLal ion (7) is actua.lly a prediction model fo r (I) with pr ~ didion horizo n d i for th e ith o utput-.
and the output Yi is only atf,~cted by t.he input. u(O) d i steps lat.f'f:
Using d w proper ddin i tion of the del ay or ders we may compute fo r Exampl f> 3.1 d 1 = cia = I , d"1. = 3 and t.he representa tion (7) takes the foll o wing fo rm
We shall modify the definition of right invertibility according t,., the above ob:scrvat ions a.nd introduce the notion!)f (d, .... , dp)-forwa rd time shift(FTS) right invert.ibility around an equilibrium point (u O, yO) of thp. RNS ( I).
YIII
+ I) = "I(t) + u, (t -
1)[",(t - 2)-
",It - :l)YI{t - 2 ) + 1/,,(1- 1)] 11, (1 + :1) = -u ,(t)u,(t) - 1,., (1)"0(1 - 1) +"3(1)]
[u,lt -
2) - v , (1 -
:l)ydt - 2) +
such that fliHlI ;r(O) E X n , loe aTe able 10 find /01' any
y,11 - I)] y,(I
+ I) = -",(I) -
4. THr,
sequ e. nce {YreJU); 0 ::; t s; t'l- } E yO a control seque. f).C(' (tic'/(t) ; 0 :0 1 :0 tF} E U I (not n , c
"3(t - l)y, 11 - 1) .
u,,'
C()NC~;PT
OF FORWAR.D TIME-SHIFT R.lGIlT INVERTIBILITY
It. is natura l to say that t.hf' system
~
i!)
ri~ht
invertible, if the map .E is sllrjective , or equival(>ntly, if there exists anothN s)'st ern ~R I : S( RP) ~.... S'( HilI) , called th(;~ right iIlvers e ~ such that the ilJput.-OIll.put ('nap of the composition of 1:/1 1 and [ IS tll(' identity map lp:
L
0
EIi I
= I"
Definition4.1 (d l , • •• , dp)-FTS right invertihility. The RNS (I) IS railed loca llv (d " .. . , dp)-forward /i1ll' !jltijl. ,-ighi int'ertihie In a tu ighb01lrhood of its equ·d ib. Mum powl (u O, yO) if then , ~xist seb U O J y l1 and XO
: S ( RI') ,- S(U").
Denot.e hy y~ the set of sequellces (Yi(t); 0 <: t <: IF} E Thell t.ht> above definitioll says that for the -ith output. component it is po~jble to reproduce locally all sequences Y n;J,i frolIl h~gil,"ing from time instant d i . But (d" .. . ,dp)-FTS rig ht invertibility does not all!)w us to rcprodur.1' the first dj t.errn:s in t h e arbitra.ry se-
Y?'
Y?_,
quell ('.e (Y'· ~ f,dt);O
S
t ~
tF]
E
yp.
5. NECr,S AHY AND SUFFICIENT CONDITIONS FOlt FORWARD TIME-SHIFT R.IGHT INV8RTIBlLlTY
If I:: b invcr Li1k iu th ~' above sellse , then it is possibit' to reproduce all arbitrary p-dil1leIlsional sequen ce {Yre/(t );O ~ t ::; I,.,) as all o ul.put, of E by ma.nipulating ~h t! input. sequence . This ddlllit iull is certainly too restri c ~iv e for most systemH ami olwiously useless for strictly oU lsal RNS or Ult' forIll ( I) . where the Hlap F doe-.s not depend nIl I/·U) . Suc:h sY5t.elll!:! cannot be right. 0 invert.ib le ill th e ah t..lv t..· sense s in Ct·, I,Jl e o utput y at. t is not, affected by I.lu: input a.llet is com pl etel y defin ed by the initial (' onditiow:, :.1:(0). ill general ,the out put. may be defined comp letely by x(O ) also at a few next. time instance8 l 1, '2, .... rI, - 1. Therefore , for t.hose systems it, is usdes.-.; to requirl~ that all S( 'qu"II f.t':-; cou ld be r ep r oducibk a.nd the bt'st we call i.u:hiev(· is (.hat aB sequences co uld Iw r ~'v rod n ('jbl t> ht'g inlliuJ;!; frorn time instant l d. For examplf>, in ('asl~ o f sy:-;tellJ (1) having delay ord ers d l ... . ,d}, we hav!' for i I, ... ,p:
From thE~ ckfinition of the di'5 the rows of th e mat.rix J((X , ll) rue nonzero vector fllucti oJls aro und (HO , VU). It is obviolls that the ran k of j((x , u) is, in general , input and o utpu t depelldent . However we shall ass ume LlJat /\(x , u) has a (·.OJls tant ran" around (UO, yO). This assumpt.ioH is formali zed in tht" not.ion of regularity of all equilibrium point.
y,ld, - I) = r:i,-l(x(O))
Definition 5.1 R egularity of an equilibriumpoiut. Wc OJII the equilibrium point (nU , yU) of the .o;y...tP. nt ( I) 1'Cy-u/al' wilh 7'e,o;prcl to (d 1• . . : dp}-FTS right in vertibility, if th e ranI.: of Ih e dccour,iif!9 matrix [{(x, u) of thf' system (1) i,~ coustant ar'Ou11d (u l\ yO).
Consider t h<' RNS (l) with delay o rd ers d i < 00, i = ., ... , p, i . ~ . the system . described by equatiolli'l (7). vVe int.T od uc ~ the so- calh.'d d en.upling matrix K(x , u) for the system (l) in t he fo llowi ng way
=
=
=
=
I
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Theorenl 5 .2 A ssum e that Jor lil t' .system (I) d; < 00, ;= I , ... ,p. Th.nlh e RN8(J)l.siocally (d 1 , ... , dp ) f orward tim e-shift right irltlerfibl e a1'Oun d a regula1' equilib rium, potnt (u u, .rl) iftlll d only ifmnk K(:r.°,u U) =p. Pr·oof. :·iuffictertc y. C onsider t,he :o;ystc m of equations (7) . By t he defill ition of the eq uilibriurll po int we have yD == A(xo, uU) . O b:O;f':rv~ that the Jacohia n ma.trixoft.be right ha nd tilde of (7) with resped 1,0 tlw co ntrol u equals to the decoupling ma.trix 1\· (J~, H). Ry the assumption of
th e theorem t.h e rank of th e decoupling matrix f{( x, u) i8 (~qlJal to JJ al. tile equilibriullI p o int. (xo, UO). SO, aft.er I\, possibl e' r~o rd f"rill g o f llJ(' c:nntrol ('.ompollellt.s we may ap ply t.h(> Irnrli cit Fund.Lon T heorem in o rder t o solve the ~y~t. e rn of equa tiom; (7) with respe ct. t o u! = 0 ( 1' 1, ... , ,u jl)T uniquely a roulld (.x . 'u 'llJeflUe u 2 = (-u ,,+ I , . .. . u lII )'I'. Th e lmpli cit. Functio n TI ..,'Orem . s ays th at i n sonle (JJos..,ibl(~ small) nei g hb o urhood (X O, UO, y U) th r:-re exists a. smoot h fUJl (" tio ll 'P of vtl riables x(t) , YI (t + d l l , . ,y,,(1 + dr ). alld u :t(f) Lt' .
whi cll if; s uch that. .p(xo, ylJ 1 u :W ) = u I II and (Yt (t+d l ), • :~p(t + rip))'" '" A(x(l), ,,(,:( 1) , YI (I + dd .. .. , yp(t + rip), . ' (t)), u'(I» . Npc('.ssity. Suppose t ha t. f.h f' system ( I) is locall y (d l ; ... ) dp}- FTS righ t iIlV(~rt i bl e around its regular equilib rium point (<<0 • .ll) . This implies , in jJuffkll~ lltly d ost'" 1.0 y:'- that. is t.h t'! fo llowing holds f id'(x( Oj, ","1) =
!/' ,/,i , i
= I , ... , p .
Assu m e thai, ran k r\"(x o, 'UO) =: k < p . A$ by reg ularity of (uU, yU) , k is con5tallt. in so me ul.' i~hb o urhood of (uO .l), tht' ra nk }\'( x , u ) in thj s ucip; hho urll ood is less than 1'. This implies that tll P fun ct.ions Fid ' (J!(O),u rc/)) , i = I , ... , p of u r ef a.rc fUll cti o nally dt'~p~I HI (:' ut , t.h at is t.here exists t he IIl ap El( ·) such I,hat il( V~', . .. , F;' , ,,(0» = R(Y" I ,j, · .. , Y"J ,P' x(O)) = O. Th" last equality mean. that .'Ire} is not. ar bit.rary but. sati s fh~ the equation H(Yrc/ ,l :···, Yr eJoi" £(0)) :::: 0 \\'h ich .;ives contradict.ioll . Th is com plf' tf's I.he proof.
=
p reql1ir(~~ m 2: p. So, p:::; m Clea rly , rank [«(;1:0 , uo) i:; a lways a nec.c::;sa.ry co ndition fol' system t o have a (d l , ... > d,,)- FTS righ t. InverSt·, t. hat, is t he system must have a t I('ast. as ma ny in ptlL:o; .:1.<0; n lll,p u t.S.
We s ho ul d like to st res... t.hat I he ass ump t ion of t he reg ularit.y of I.he l~qllilihri um poiul. (xO , UO) in Theorem is extremely vital. If th e point (uY, yO) is not regu lar , th at is around t ht' point (u O, yo) the rank of t he deco1lpling mat.ri x I«x , u) i5 f10 1. Tlr:cessarily cons tant: t he n t he condi tion Jf (XO , uO) == /, is not neC~1)8ary fo r (d I, ... , dp ) - FTS right invertibilit.y. The illustriLt ion of th is phenom en on is giveu ill t h e follmving simple exam ple: y(t) = u(t - l ):i. We have /\(x , 1/) i)A (x , .d/ llu = 3,,' . At th e equilibrium poi m uO 0 , yO = 0 the rank of [«(x, u) is equal to 0 whi ch is less t h a n p I. Still. th e arbitrary S(~ q uen Cf:S a re reproducible for I :S t :S tp by the choice of contro l ut i) Vy(t + I) . The reasoll i. that. the POilll UO = f) t = 0 is not a reg ula.r equ ilibri um point . Tht, rank of the mat.ri x If(x , u) lS equa.l t o l at. aH points
=
l'
,,;t'
=
=
=
o.
From equations (9 ) it is dear that there exist. no rig ht inverse for a strict ly ('.allsal recursive sysLem E such that t.he input 1.0 th e orip;inal systl!m can b e co lClp uted as t,h,' output of th e in verse syst.e.n . wit hout u!)in g t.he fllt.lln · values of I.he refercnc' t' signa l. Act ually, for «(1 1 , ... , rlp ) FTS right. inw,rt.i ble s ystem ~, t he out put. of the right. ill verse system (i.e. the cont rol of the origillal system ) at tim e ins ta nt I will d cp~ncl o n the ith compo nent the refere nce: output at. tinw ins tant t + dj. At; Y r ej i~ generat.ed by the de!)igller , ill th e actual control law de!:iign t hi s implies t hat a change of reference signal must, he. pr.~p l ann e d so m ~ t im ~ s tt·ps a head , whi ch is oft.en a realisti c. ass umpt.i on. When it is possib h', the right. invertie s Y l!i t,( ~m can be realized
or
6. EXAMPLES
Exanlp le 6.1 Consider the RNS
= 1.11(/ -
+ " , (I - 2)y,(1 - I) y, (t ) = 1.1,(1 - 2)[lI ,(t - 2) + IJ
YI (t)
I)
under t he iuitial c.on ditions [UT ( -I , - 2) , ~'T ( - 1, - 2)rl' for I ~ O. Tlw delay order,; of this system are d 1 == 1, d'J, ::: 2 an d t.he system can be rc present,f':d in th e fo rnl YI (I
y,(1
+ I) = "I (t) + U2( t - I )ll'( t - 2)[Yl (I + 2) = U2( t )[UI(1 - I ) + u,( I- 2)y, (t -
2)
I)
+ I] + IJ.
This systcln is ( 1, 2)-FTS right in ver tible a ro und an equilibrium puint wit.h :j:. -1 . The equatio ns of th e right. Inverse are:
y?
2410
12
11,( I) = y, (I + 2) 1[1 + .." (t - I) + u, (I - 2)y,( t - 1)1· Exarnpl" 6.2 ConsIder the R fI'S
= ",(I id/) = " I (I. -
VI(!)
3) -'''t(t _. 2)
2)/[y;(t - 2)
+ I]
und er the in it ial condit.ious [l.' T (- I, - 2) , U'/'( -I , _ 3)JT for t ~ O. The delrLY orders of this sjlstem are d1 = <1'2 2 and so Llw system can be fcpn~~;t' n1. e d in the form
=
+ 2) =",(1 - I) _. "I (/) y,(t + 2) = ut(l)/[lI ~ (I - 3)- 2«,(t +u;U - 2) + I ] .
YI (t
via th e independent. ones and apply to t.he dep elldent. equatioll~ the one-step forward t im e-shift. operator and repeat the whole. procedure (:;ay 0: times) until we obtaill a syst~ml of eq uations which can be solved for the ('ontrol u( I) in t.e rms of x( I) and y( t + I), y( t + 2), ... , y(t + a) in case of arhitrary referell ce signal ) o r it will be-come dear that the lat.ter is impossible If it j~ possible to ob tain a. systp.l1l of equatiou!) which can be solved for t he COI)trol t.hen we are a ble to rcprorlu ce at the ilh output Yt 3n a.rbitrary reference signal starting from certain tirrlt' inst.ant Cf, 2" d i with (1'1 > d i for some j E {I ) ... , p). The gellf'raliilat.ions a lnng t.ht'se lines will be the t.opir. of another pa per.
3)ul (I - 2)
It is c1en..r t.hat this syst.em is Hot lo ca lly (2, 2)-FTS right invertible. since the rank of th e IIlatrix f{ is equal to one fo r all po:isiLk t'lJuilibrium poilll,s. Howt"ver, the arbitrary refe remT s ign <'l.ls can bl ~ g(~ lI e r at.ed at the fi r!;t ouLput start.ing frOHl. time insta.nt ;1 and a.t the s~con d output. start ing from t.ime instant "2 hy the choice of th e followin g w Hlml
8. RJ::FERENCES
Bastin (;., F. Jarac"i and LM.Y. Mareels (1993). Dead beat cont rol of rec ursivl~ lloIJliuear syst.ems. Prof:. :1.2nd Con/. Qn Derision and Conlrvl, San Ant.onio.
Tex".,
2965- ~971.
El A.Ill; S. and M.Fli ess (1992). Invert ibilit.y of discretet.imt~ systems. Proc. 2nd IFAG' Symp. on I\To nJinea r ControI5'y8lf'lIts Desig n: Bordeaux, 192 196. Fliess, M.(l990, . Aut.omatiqueen temps discrd ct a lgcbre aux differcnr.f'S. Forum Malhemalicuflt ) 2 ) 21~~·
2:11. Grizzle, J.W. (1993) . A linear algebraic framework
7. DlSCllSSION A!\D (:ONCLUSIONS
For systen l L.; defilLCJ by ( I) no possibility exist to reproduce at th(: output an arbitrary reference signal starting from timt': iust.a"t t = O. We are a.ble to reproduce the refereno' ~ i g nals at tilt> outpu t wiUt some time-shifts and the :;rnall('~s t, possibk valu(~ of till' l.in\ ~ - shift is di (the delay orciN) for i1.h o ut.puL comp(Jn l:f Jt.. These smallest vrtlllt"s (" ll t be reali£' ~ J if Uw :-;y!"t.f'tll o f .;oquations
can be solved for u(ti for arbitrary [Yi(1 +,It}, ... , Yp(t + dp )}'I' . Note that \VI ' fa. nnot solve t l ll.~ ::;ystem of equa..tions (10) fur u(l) ill c:a~t': of arbitra.ry left hand side if some rom poncnts of the vector funrt,i 0 11 .4(;c, u L as fll netiOIlS of th e control. d ~pelld fUllctiollally on the oth ers , or equiva)PIlt.iy, if rank ;~A(x. u)/iJIt < p .
for t. he analysis of discrde-time nonlinear systems. SIAM J. COlltr. and Op/m,iz., 31,1026-1044 . Halllnw r.l . (19x4) . Nonlinear sy~tcrns: sta.bility and rationality. 1nl. .f. Conir., v.40,1 - j5. Kot ta fr. (19S6). On the iuverse of discrete-tirrw lin ear-analytic system . Control- Theory and Ad-
van"cd Tcclwology, v. 2, 619- 625. Kotta, i"J. (1990). On the right inverst:' of discretc-t.imc nonlinear syste.ms. tnl . .I. Contr. , 51 : (-9. Leontaritis LT. and S.A. Billiugs (1985). Input output para.met.ric models for non-linear systems. ParI. I : deterministic lIonlilJe:u systemf';' Tnt. J. C01,t1'., vAI. :1O:1-32~. Monaco S. anti D.No rmallll-Cyro t. (1987). Minimumph a...:;e nonlirwa r disc.rete-t.ime systems and feedback stabili~ation. Pror:. 26t1r IEEE Conf. on DC8~ ,<;ion an.d Control, Los Angelfs, CA , 979 · 986. Morh a('. M. (IH90). Deterllltna.tion of inverse Vo Jt~ rr
Nijnwij f'J> H. (1989) . On dynami c dccoupling and dynamic path cont.ro llahility in economic systems. ,I OUT"lIaf of Emlwmi,: J)yna mics and Control, v . l3 ,
11 :19 . The idea [,0 gellt~rali ze the notion of right invertibility is to rcpresf'nt. the fun c.t ionally depcndr:nt. components
2411
Copyright © 1996 [FAC 11th Triennial World Congress, San Francisco. USA
ON RIGHT INVERTIBILITY OF NONLINEAR RECURSIVE SYSTEMS VUc Kotta'
• 11lst.itute of
(.'yb e rn e ti(:.~ .
Akadeemia t ee!!1, Tall1. nn. 0026,
Es to1~ia .
F-mail:
kotla @ioc.e.e
Abstract.. The right. invertibility pro blem is st.udied fo r a daso of rec ursiv 4~ lion linear systems (RNS), i.e . for systems, modelled by rec Llrsiv(~ nonlill t~ar input-output equat.ions invo lving o nly a finite number of input values and a finite numb er of o utput values. Th e con c.ept of delay orders and the spec ial ease o f right inverl ibility, the notio n of (d J, . , J II )-fo rwarJ time shift (FTS) right invertibility, kn own rur discretet im e nonliu ear systems in state space form : are extend ed t.o R NS. Necessary and s uffici e ll t condit.ion s fo r ln cal (d 1 , .. ' Jdp)-FTS right invert.ibility are propost ~ d. Finally, il: is s hown ho w 1.0 gt 'nf: ralize the :;pecial case of right invertibility - (d. , . . ,
Keywords. ltol1l illCiH f t";cursivt' system, in ver1.ibility. inverse sp;tem
\. INTRODU CT ION The fundam ental ques tion s of tl1 e 4 ~ xistence and the COIlst. ructi o n of righl. in\'~~ rses for uonlinear discrcte- time d ynamical system!> art' addr€:,.l:)scd ill thi s pap er. Such a
right. iuvNse is intuit.ively und nrs tood to he a. second nonlin ear sys t em Sll ch th at. whfm tlw original system is appli ed in seri es wit.h this rig ht inverse, t hen its outp ut.s are equ a l t.o tli (' inputs of t.he right iu verse system. B~
ca use of' the illherent. delay typi cally found in dynamical sys tems , in it. great. Ilumbf:"'r of cases 110 s uc.h invers io n is p ossibl t', and tht~ pwhlt~1Tl is of liJllited inteH~s t. . Greater g<'ll e rali ty is o hta in NI hy cO Jl s id ~ rifl g a notioll of forw a rd t.ifTIt' shift right illv!,~ r ~c ill which 1.114' input to the r ight inverse l:)ysi:elll is not j list t.ht' refe rl' IKt' signal Yr eJ (t ) but the fefr: r(' llce signal at some fut,ure time inst a nt t + (r, i .c. Yr l'! / (I, + n ). The determinat.i on of th e smallest possihie valu e for rr is a questi o n of prdcl-ical and th cor ~t.ical import a nce. In cas(" of Illlllt.i--outpul sys t.ems , the value of I.hl:' s mallest possihk 0 , ill ~f> lI e ra l, is not t.he sanw
di sc r~t e - tirne
systern, delay orders, right
for all components of the rt fcrence sign al. A very impor tant concept in treating sys tem inversion from f.his generaliz-ed point of viow, is 1,he COll ccpt of delay ord(~ r s.
Except t.h e pap"r (Morhac . 1990) on finding the in\:erse Vo lterra represent atio n , most pre vjo us work 0 11 this snbj ed ha.':' r.oncentrated on sys tems h
19 90; Grizzle, 1993; Kotia, 1986; Kot.ta, 1990; Monaco and Norruand- Cy rot , 1987 ; Nijrneij er, ] 989). The purpose of our paper is to s tudy the fo rward t.im~ s hift right
invertibili t y pro blem for a dass of systems described by r e:c uf~ iv(~ no nlillear input ·o ut.pu t equ a tions illvol v in g only a fillit,r' number of i IIplll valu e~ and a finite IlUmb e>f of o utput. values. Such sys te ms are called recursive n Oli lint.:a r syslems ( R:"lS) (Ha[lltncr, 1984; Leontaritis a n d Billings , 198.5) . \/lle sha ll f! xte nd Lo this class of nonl in· car sy stl ~ m s t.he conce j.>l of delay orders :md the notio n of (d J , • . ,dp ) - FTS ri ~ ht invertibility. The latt.er is a sp ecial case of F'l'S right iJ1ve rtibility in c.ase of whi ch
2406
th e smallest poss ible values for (1 ) ' . , Up may be rea lized. Note t.hat in (BMt in et al. , 1993) for single-input single-output RNS the qu estion of def\.d- beat control has been ~t udit~d . Th e latter prob lem is closely related t o the right inwrtibility problem . Howpver , unlike (Rastin el Ill., 199:1) W~ consider I..llt' rnull.i -input m ult.i-outpu t !;yS 1. ~ ms and CUllcentrat,€,: Ot] t.he lo('.a l solutions around an equilibrium point. of t h ~ system .
equality yO = P( y O, Un) wh"'e y o = (yO ,7' , . . _, yO ,T)"" UO = (u.0 ,T, ... , uO .T)T.
From n ow on, we co nsider RNS (1) at. nonn egative tim e steps in a. finite tim e interval 0 ~ t ~ tF under iui t ia l conditio ns
Denote by
2. DESCRIPTION OF RECURSIVE NONLINEAR SYSTEM In th is secti on , b e:-;idl~s recalling Uw notion of rec ursive nonlinear sYf:item (Eal5tin et al. , ) 993; Hammer, 1984), wc estab lis h so m l~ no tatio ns and inl,fod w:.e some pre liminary mat.l'ria.L We denot.e by 5'( Rm) t.h e sl2:'l. of fLU two-sided infinit.e sequ ences of the fOrlll
{ =(t)} = (
, =( - I ), =(0 ), .,0), z(2) , .. .)
where z(l ) E R:n for all integer:',. l. A dynamical syst.e m i, a map ~ : scum) ~ 8( 81' ) ("(i)l ~ (yet )) which tr ansforms input seCJIl ~nr;~ {u(l)} illl, u output sequence {v(l)). Given two sys!.e rn s ~, S(lI"' ) ~ 8(RP) and E,: S(81') ~ S(Hq) , denot(' by :S', 0 E, : 8( R'" ) ~ S( R.9) t.he sy ste lTI represen Le d hy t. he ('omp o!:!ite map . A finit.f' suh seq uellce of t, hf: infinik :-;equencf' {z(t)} be-twcen tinw i nsta ncc~, t a nd t - T SI,H',ked in t.h ~ co lumn vecto r is deno ted by
w.
.,,(1) '"
=
p(y . - I ( 11.- 1.) 1-}_'
I- v
(I)
w he re F Rl l l' + lIfH I--T RP is all an a lytic map and I < Jl S; :::xJ: 1 S v ::; 00, then t he syste-m 1: is said t.o have a 01l115al finite di ull:' llsion al rea lizatiun .
D efinition 2 .1 R ecursive llonlillear system . Recursive non/iu ra1' sY8 tcm ( R NS) v; It ·""Y8 tem which I,as a cflllfi ul fimL i' diwou;iona[ rea lizatwlI oJ the form (1). D efillitiulJ 2.2 Equilibriunl point . '/'he pair' of conslunt
TJu{tUS
(tl°, yO) is ca,lIed th e equ'ilibritnn powl of tht (IL l! , ',!l ) Batisfies the
a (J.l.p + vm )-dir ne nsional vector (yO,T , . the system (1) has as inputs
t he ''''tu pnce u = {u(t);O ~ I ~ tF}. Throughout the pa per we sh a ll adopt a local viewp oin t. More pf(~ c isely , wc work aro und an equilibrium p oillt (u",yo) of the syste m (1) , l et u, deno te hy U O (resp . U) the sd of r.o nlfol sequell'es " {u(t ): 0 ~ t ~ t F) (resp. Utt~;) sucb tha.~ the cllntrols u(t ) fo r every tare sufficien!.ly close to u· , i,e. that 11 u(t)-u· I I ~ 6 fur som e fi > 0, Allalogously , let us ,knote by y. (resp . Y) the se t. of 011 t PIIt. sequellr<.'S (yet ); 0 ~; I ~ I,,} (resp. )','~"' ) s uch tbat the o utput. ~ y(t) for eve])" t are sufficie ntly close \.0 yO, i.e . that 11 yll) - yG 11< , fo r sorne, > O. Finall y, let us de not.I ' XO t he ne ighbq urhot:1 of ;r;o s uch tha~ fo r every x E X ", 1I :r - x" 11< -; fo r some -y > O.
=
uy
For difference equation (1) Illld ~r initial conditions x(O) , as long as F is a wdl-dcfinf'd function of /l..,",P + VIn , t,hP.ft" is no problem rcga.rd ing f!xi stence and uni4ueness of itl' solution y(t; 0 ::; t :5 t.p), for a rbitra.ry control sequenc(' u E 11 0 anrl arbitrary initial (ond ition x(O) E XO. Such Cl solul,ion will be df' noted as yH 1 x(O) : u ) which is a sho rlhand writing for !I(t , x(O), u( O), . . _.u(t - I)).
Z:_r
If T ~ 0 , it. is unders t,ood thai, denotes a n empty subsequen ce of {~( t)).lffor e very in put sequen ce (u (t)] , th e corre~poncling output ~e4l1cncr {y(t)} of th e ~y~tcm E sa.tisfies t he equation
:1;0
yO,T I .(to ,T : ... 1 uO ,T)T . Then
3. THE DELAY ORDERS WITH RESPECT TO THE CO\'TROL For discrete-time Ilo nlinf!ar ~ys t em s J describ ed by ~t.at(' equatioJl)), t he dd ay o rders d" i 1, .. . , 1' , (referred also in the literature as the char;Ldcristic num bers, the relative orders or the dead tilIll ~S ) with resp ect to the (.o ntrol h avl' been defined , one fQr ~ acb output compon ellt.. These system stru ct ura l para meters tell us how m a ny inherent delays tl lC re a re bet wee n the ith tornpo nent. Yi of the o utput and t.lw contr'.>l , or equiva lent.ly, for how many fir:o:;t time ill stances Yi jf:) complet.e1 y defined by t hl' initial cundit.ions a.nd whi ch is t he firs t tim e instant fo r which th e potiSi bility arises t.o change Yi a rbitrarily. tll this secti on we sh a ll extend ~h e concept of delay orders to the dass of R NS . Definf~ t h t:> it h ,-omp onent of Jo' ill (I) hy F;.
=
7·t f' u rs ive llouli n(~(I1' ,'iys/ em (I) if
2407
At. firsL sight it. m a.y seem t.hal. ont." c:w defin~ di as t.he !o;mallest. p ositive integer k such (.li at ')
(
i)t((l - k)
F(y' - I /1' -1 ) , I - I" I-v
tt,,,
is not identically Zf'ro. Using t.he abo\'l~ definitio n, a RNS (1) wit.h del ay orders d l , ... , d p admits for i l, ... , p a representa t,ioll o f I.he torm
=
y,(I
+ d,) = F; (Yc':t ~; , !t (t),
[I,'+1._v) '
(2)
However , th ~ above defi.nit.ion is noL , in general , in co mplete correspondence with the staLe spaw formulat ion of tJlf' concept. sinct~ Lt. dn e~ not ddhw rUI' how many firsL rirJ'J E' insLann;s Yi is cornpleLcl y defin ed by the initial condit,ioJls ohserve that (2) co ntain..., y (l +d i - I ), ... , yet) whi ch are not the part of illitial C{ .nditions nnd may depend 011 co ntrol. Th l~ full owin g f'xalllplc serv ~ s as an illus trat.ion.
Example 3.1 Crm.'lfder th e ·<;Y5I r m
= "l(t y,(I) = ",(I. -
1/,(t)
I)
+ ",et -
2) -
=
Yi (l
U3(t --
yet + 2) = F/(y(t), Y'-~;+I' u(t), U,':;+I)
= Fl(Y.'~:, l ',':,~, u(I)),
I)
If this vec lor .i~ non zero on all open and dense subset. of Y x 0 , we set di 2; oth' ~rwise we continue with
2)y" (i - 2).
=
l(t Yz
H ow ~~ ver,
actu a.lIy there possihility to c.hauge y'.!(t I- 2) a.rb it rarily sinc;e (4 )
+ 2)
= V,' (yI '- -I U'-I) IJ' t-v'
+ 2) =
(Jj
(6)
In this way Lhe numb~r di - If it e xist. ~ - determines the inherent delay bd.ween the i1lputs and th ~ it.h output.. A R NS ( I) with delay ordel's d" i represent.ation of the form
=
I, ." , P admit. a
and if we replace Y3(t. + I ) in (j) by I,he RHS of (4) ill (:3) we see that Y2(t + 2) depends ('.olllp letely 011 t he initia l c.ond itions· Y2(l
(5)
2)!J2(t ·- I)
By the abovf> definiti on, d 2 = L 11 0
,,'.-'1/ ) • = Fi ' (Y'-' t-Il ' v,
Apply aga ill a. forw ard shift operat.or to equation (5) a lld replace in the lat ter yet) via the right hand ,ide of (1):
Co mpute
is
+ I).
COfnput.t' in an a ll a logo us fa..... hion the derivative
",(I - :l)YI (I .- 2) + Y3(1 -
y,,(t) = - ,.. , (I - I) -
From the analytirity of the >y.tem (I) it follows that either vector dF/ Of du( I) is non zero for all (Y.'~~ , U:~:) belonging to an open alld dense subset OJ of Y X 0 or this v(~cto r vanishef:> fol' all (Y/_-t41 , Utt~~) E 1;' X U. In the fir::d. ra-se W f" define di J whereas in the latter case we cOHt.inu(' by observing tlwt the fun ctio n J/~t does uot. depend o n u(t) , i.e depends completely on the in it.ial conditio.tls and so we may writ ~
(7)
-1(,(t - I)YI (I) - ",(t - l)y, (t - I ), o r ill thE' vec tor form
N ~x t
wc shall give the propN ddlllit.ioIl of t he delay ord e r~ for RNS. The forward ti11l e shift operato r 6 is denll ('(1 as 6y(l) = y(l + I) . Apply t he o llc-slep forward ~ "ift. operal,or 1.0 etlllatio n (I) a.nd f._~ place in the latter y(t) .... ia the initial conditions, i.4 '. via t he right hand side of (1) in orciN to obtain
y( I
+
YI(t +, [
y,(1
dJl]
= .4.(x(t), u(t)).
(H)
+ dp )
where
I) = F(y(t) , 1'','_-':+1 ' ,,( I), [!,'~ ~+ I)
= f 'I(Yt-' - !"'
(.'-I
'I-v''//.
Denote 1.lU' ith componem of /0'1 hy i = 1, .. .. p the d~rivati vt'
It )) .
P/
By definition of (Dastin et al., 1993) a prediction m odel and compute for
with predi ction ho ri zon d for a R NS is a rc("ursive map which a llo ws t.he computati on of thp. output at futur e
2408
tim e inst.ants t + d from inputs up to I.jJn~ t and outputs IlP to time t - 1. According 1.0 the a.bove definition the represenLal ion (7) is actua.lly a prediction model fo r (I) with pr ~ didion horizo n d i for th e ith o utput-.
and the output Yi is only atf,~cted by t.he input. u(O) d i steps lat.f'f:
Using d w proper ddin i tion of the del ay or ders we may compute fo r Exampl f> 3.1 d 1 = cia = I , d"1. = 3 and t.he representa tion (7) takes the foll o wing fo rm
We shall modify the definition of right invertibility according t,., the above ob:scrvat ions a.nd introduce the notion!)f (d, .... , dp)-forwa rd time shift(FTS) right invert.ibility around an equilibrium point (u O, yO) of thp. RNS ( I).
YIII
+ I) = "I(t) + u, (t -
1)[",(t - 2)-
",It - :l)YI{t - 2 ) + 1/,,(1- 1)] 11, (1 + :1) = -u ,(t)u,(t) - 1,., (1)"0(1 - 1) +"3(1)]
[u,lt -
2) - v , (1 -
:l)ydt - 2) +
such that fliHlI ;r(O) E X n , loe aTe able 10 find /01' any
y,11 - I)] y,(I
+ I) = -",(I) -
4. THr,
sequ e. nce {YreJU); 0 ::; t s; t'l- } E yO a control seque. f).C(' (tic'/(t) ; 0 :0 1 :0 tF} E U I (not n , c
"3(t - l)y, 11 - 1) .
u,,'
C()NC~;PT
OF FORWAR.D TIME-SHIFT R.lGIlT INVERTIBILITY
It. is natura l to say that t.hf' system
~
i!)
ri~ht
invertible, if the map .E is sllrjective , or equival(>ntly, if there exists anothN s)'st ern ~R I : S( RP) ~.... S'( HilI) , called th(;~ right iIlvers e ~ such that the ilJput.-OIll.put ('nap of the composition of 1:/1 1 and [ IS tll(' identity map lp:
L
0
EIi I
= I"
Definition4.1 (d l , • •• , dp)-FTS right invertihility. The RNS (I) IS railed loca llv (d " .. . , dp)-forward /i1ll' !jltijl. ,-ighi int'ertihie In a tu ighb01lrhood of its equ·d ib. Mum powl (u O, yO) if then , ~xist seb U O J y l1 and XO
: S ( RI') ,- S(U").
Denot.e hy y~ the set of sequellces (Yi(t); 0 <: t <: IF} E Thell t.ht> above definitioll says that for the -ith output. component it is po~jble to reproduce locally all sequences Y n;J,i frolIl h~gil,"ing from time instant d i . But (d" .. . ,dp)-FTS rig ht invertibility does not all!)w us to rcprodur.1' the first dj t.errn:s in t h e arbitra.ry se-
Y?'
Y?_,
quell ('.e (Y'· ~ f,dt);O
S
t ~
tF]
E
yp.
5. NECr,S AHY AND SUFFICIENT CONDITIONS FOlt FORWARD TIME-SHIFT R.IGHT INV8RTIBlLlTY
If I:: b invcr Li1k iu th ~' above sellse , then it is possibit' to reproduce all arbitrary p-dil1leIlsional sequen ce {Yre/(t );O ~ t ::; I,.,) as all o ul.put, of E by ma.nipulating ~h t! input. sequence . This ddlllit iull is certainly too restri c ~iv e for most systemH ami olwiously useless for strictly oU lsal RNS or Ult' forIll ( I) . where the Hlap F doe-.s not depend nIl I/·U) . Suc:h sY5t.elll!:! cannot be right. 0 invert.ib le ill th e ah t..lv t..· sense s in Ct·, I,Jl e o utput y at. t is not, affected by I.lu: input a.llet is com pl etel y defin ed by the initial (' onditiow:, :.1:(0). ill general ,the out put. may be defined comp letely by x(O ) also at a few next. time instance8 l 1, '2, .... rI, - 1. Therefore , for t.hose systems it, is usdes.-.; to requirl~ that all S( 'qu"II f.t':-; cou ld be r ep r oducibk a.nd the bt'st we call i.u:hiev(· is (.hat aB sequences co uld Iw r ~'v rod n ('jbl t> ht'g inlliuJ;!; frorn time instant l d. For examplf>, in ('asl~ o f sy:-;tellJ (1) having delay ord ers d l ... . ,d}, we hav!' for i I, ... ,p:
From thE~ ckfinition of the di'5 the rows of th e mat.rix J((X , ll) rue nonzero vector fllucti oJls aro und (HO , VU). It is obviolls that the ran k of j((x , u) is, in general , input and o utpu t depelldent . However we shall ass ume LlJat /\(x , u) has a (·.OJls tant ran" around (UO, yO). This assumpt.ioH is formali zed in tht" not.ion of regularity of all equilibrium point.
y,ld, - I) = r:i,-l(x(O))
Definition 5.1 R egularity of an equilibriumpoiut. Wc OJII the equilibrium point (nU , yU) of the .o;y...tP. nt ( I) 1'Cy-u/al' wilh 7'e,o;prcl to (d 1• . . : dp}-FTS right in vertibility, if th e ranI.: of Ih e dccour,iif!9 matrix [{(x, u) of thf' system (1) i,~ coustant ar'Ou11d (u l\ yO).
Consider t h<' RNS (l) with delay o rd ers d i < 00, i = ., ... , p, i . ~ . the system . described by equatiolli'l (7). vVe int.T od uc ~ the so- calh.'d d en.upling matrix K(x , u) for the system (l) in t he fo llowi ng way
=
=
=
=
I
2409
Theorenl 5 .2 A ssum e that Jor lil t' .system (I) d; < 00, ;= I , ... ,p. Th.nlh e RN8(J)l.siocally (d 1 , ... , dp ) f orward tim e-shift right irltlerfibl e a1'Oun d a regula1' equilib rium, potnt (u u, .rl) iftlll d only ifmnk K(:r.°,u U) =p. Pr·oof. :·iuffictertc y. C onsider t,he :o;ystc m of equations (7) . By t he defill ition of the eq uilibriurll po int we have yD == A(xo, uU) . O b:O;f':rv~ that the Jacohia n ma.trixoft.be right ha nd tilde of (7) with resped 1,0 tlw co ntrol u equals to the decoupling ma.trix 1\· (J~, H). Ry the assumption of
th e theorem t.h e rank of th e decoupling matrix f{( x, u) i8 (~qlJal to JJ al. tile equilibriullI p o int. (xo, UO). SO, aft.er I\, possibl e' r~o rd f"rill g o f llJ(' c:nntrol ('.ompollellt.s we may ap ply t.h(> Irnrli cit Fund.Lon T heorem in o rder t o solve the ~y~t. e rn of equa tiom; (7) with respe ct. t o u! = 0 ( 1' 1, ... , ,u jl)T uniquely a roulld (.x . 'u 'llJeflUe u 2 = (-u ,,+ I , . .. . u lII )'I'. Th e lmpli cit. Functio n TI ..,'Orem . s ays th at i n sonle (JJos..,ibl(~ small) nei g hb o urhood (X O, UO, y U) th r:-re exists a. smoot h fUJl (" tio ll 'P of vtl riables x(t) , YI (t + d l l , . ,y,,(1 + dr ). alld u :t(f) Lt' .
whi cll if; s uch that. .p(xo, ylJ 1 u :W ) = u I II and (Yt (t+d l ), • :~p(t + rip))'" '" A(x(l), ,,(,:( 1) , YI (I + dd .. .. , yp(t + rip), . ' (t)), u'(I» . Npc('.ssity. Suppose t ha t. f.h f' system ( I) is locall y (d l ; ... ) dp}- FTS righ t iIlV(~rt i bl e around its regular equilib rium point (<<0 • .ll) . This implies , in jJ
!/' ,/,i , i
= I , ... , p .
Assu m e thai, ran k r\"(x o, 'UO) =: k < p . A$ by reg ularity of (uU, yU) , k is con5tallt. in so me ul.' i~hb o urhood of (uO .l), tht' ra nk }\'( x , u ) in thj s ucip; hho urll ood is less than 1'. This implies that tll P fun ct.ions Fid ' (J!(O),u rc/)) , i = I , ... , p of u r ef a.rc fUll cti o nally dt'~p~I HI (:' ut , t.h at is t.here exists t he IIl ap El( ·) such I,hat il( V~', . .. , F;' , ,,(0» = R(Y" I ,j, · .. , Y"J ,P' x(O)) = O. Th" last equality mean. that .'Ire} is not. ar bit.rary but. sati s fh~ the equation H(Yrc/ ,l :···, Yr eJoi" £(0)) :::: 0 \\'h ich .;ives contradict.ioll . Th is com plf' tf's I.he proof.
=
p reql1ir(~~ m 2: p. So, p:::; m Clea rly , rank [«(;1:0 , uo) i:; a lways a nec.c::;sa.ry co ndition fol' system t o have a (d l , ... > d,,)- FTS righ t. InverSt·, t. hat, is t he system must have a t I('ast. as ma ny in ptlL:o; .:1.<0; n lll,p u t.S.
We s ho ul d like to st res... t.hat I he ass ump t ion of t he reg ularit.y of I.he l~qllilihri um poiul. (xO , UO) in Theorem is extremely vital. If th e point (uY, yO) is not regu lar , th at is around t ht' point (u O, yo) the rank of t he deco1lpling mat.ri x I«x , u) i5 f10 1. Tlr:cessarily cons tant: t he n t he condi tion Jf (XO , uO) == /, is not neC~1)8ary fo r (d I, ... , dp ) - FTS right invertibilit.y. The illustriLt ion of th is phenom en on is giveu ill t h e follmving simple exam ple: y(t) = u(t - l ):i. We have /\(x , 1/) i)A (x , .d/ llu = 3,,' . At th e equilibrium poi m uO 0 , yO = 0 the rank of [«(x, u) is equal to 0 whi ch is less t h a n p I. Still. th e arbitrary S(~ q uen Cf:S a re reproducible for I :S t :S tp by the choice of contro l ut i) Vy(t + I) . The reasoll i. that. the POilll UO = f) t = 0 is not a reg ula.r equ ilibri um point . Tht, rank of the mat.ri x If(x , u) lS equa.l t o l at. aH points
=
l'
,,;t'
=
=
=
o.
From equations (9 ) it is dear that there exist. no rig ht inverse for a strict ly ('.allsal recursive sysLem E such that t.he input 1.0 th e orip;inal systl!m can b e co lClp uted as t,h,' output of th e in verse syst.e.n . wit hout u!)in g t.he fllt.lln · values of I.he refercnc' t' signa l. Act ually, for «(1 1 , ... , rlp ) FTS right. inw,rt.i ble s ystem ~, t he out put. of the right. ill verse system (i.e. the cont rol of the origillal system ) at tim e ins ta nt I will d cp~ncl o n the ith compo nent the refere nce: output at. tinw ins tant t + dj. At; Y r ej i~ generat.ed by the de!)igller , ill th e actual control law de!:iign t hi s implies t hat a change of reference signal must, he. pr.~p l ann e d so m ~ t im ~ s tt·ps a head , whi ch is oft.en a realisti c. ass umpt.i on. When it is possib h', the right. invertie s Y l!i t,( ~m can be realized
or
6. EXAMPLES
Exanlp le 6.1 Consider the RNS
= 1.11(/ -
+ " , (I - 2)y,(1 - I) y, (t ) = 1.1,(1 - 2)[lI ,(t - 2) + IJ
YI (t)
I)
under t he iuitial c.on ditions [UT ( -I , - 2) , ~'T ( - 1, - 2)rl' for I ~ O. Tlw delay order,; of this system are d 1 == 1, d'J, ::: 2 an d t.he system can be rc present,f':d in th e fo rnl YI (I
y,(1
+ I) = "I (t) + U2( t - I )ll'( t - 2)[Yl (I + 2) = U2( t )[UI(1 - I ) + u,( I- 2)y, (t -
2)
I)
+ I] + IJ.
This systcln is ( 1, 2)-FTS right in ver tible a ro und an equilibrium puint wit.h :j:. -1 . The equatio ns of th e right. Inverse are:
y?
2410
12
11,( I) = y, (I + 2) 1[1 + .." (t - I) + u, (I - 2)y,( t - 1)1· Exarnpl" 6.2 ConsIder the R fI'S
= ",(I id/) = " I (I. -
VI(!)
3) -'''t(t _. 2)
2)/[y;(t - 2)
+ I]
und er the in it ial condit.ious [l.' T (- I, - 2) , U'/'( -I , _ 3)JT for t ~ O. The delrLY orders of this sjlstem are d1 = <1'2 2 and so Llw system can be fcpn~~;t' n1. e d in the form
=
+ 2) =",(1 - I) _. "I (/) y,(t + 2) = ut(l)/[lI ~ (I - 3)- 2«,(t +u;U - 2) + I ] .
YI (t
via th e independent. ones and apply to t.he dep elldent. equatioll~ the one-step forward t im e-shift. operator and repeat the whole. procedure (:;ay 0: times) until we obtaill a syst~ml of eq uations which can be solved for the ('ontrol u( I) in t.e rms of x( I) and y( t + I), y( t + 2), ... , y(t + a) in case of arhitrary referell ce signal ) o r it will be-come dear that the lat.ter is impossible If it j~ possible to ob tain a. systp.l1l of equatiou!) which can be solved for t he COI)trol t.hen we are a ble to rcprorlu ce at the ilh output Yt 3n a.rbitrary reference signal starting from certain tirrlt' inst.ant Cf, 2" d i with (1'1 > d i for some j E {I ) ... , p). The gellf'raliilat.ions a lnng t.ht'se lines will be the t.opir. of another pa per.
3)ul (I - 2)
It is c1en..r t.hat this syst.em is Hot lo ca lly (2, 2)-FTS right invertible. since the rank of th e IIlatrix f{ is equal to one fo r all po:isiLk t'lJuilibrium poilll,s. Howt"ver, the arbitrary refe remT s ign <'l.ls can bl ~ g(~ lI e r at.ed at the fi r!;t ouLput start.ing frOHl. time insta.nt ;1 and a.t the s~con d output. start ing from t.ime instant "2 hy the choice of th e followin g w Hlml
8. RJ::FERENCES
Bastin (;., F. Jarac"i and LM.Y. Mareels (1993). Dead beat cont rol of rec ursivl~ lloIJliuear syst.ems. Prof:. :1.2nd Con/. Qn Derision and Conlrvl, San Ant.onio.
Tex".,
2965- ~971.
El A.Ill; S. and M.Fli ess (1992). Invert ibilit.y of discretet.imt~ systems. Proc. 2nd IFAG' Symp. on I\To nJinea r ControI5'y8lf'lIts Desig n: Bordeaux, 192 196. Fliess, M.(l990, . Aut.omatiqueen temps discrd ct a lgcbre aux differcnr.f'S. Forum Malhemalicuflt ) 2 ) 21~~·
2:11. Grizzle, J.W. (1993) . A linear algebraic framework
7. DlSCllSSION A!\D (:ONCLUSIONS
For systen l L.; defilLCJ by ( I) no possibility exist to reproduce at th(: output an arbitrary reference signal starting from timt': iust.a"t t = O. We are a.ble to reproduce the refereno' ~ i g nals at tilt> outpu t wiUt some time-shifts and the :;rnall('~s t, possibk valu(~ of till' l.in\ ~ - shift is di (the delay orciN) for i1.h o ut.puL comp(Jn l:f Jt.. These smallest vrtlllt"s (" ll t be reali£' ~ J if Uw :-;y!"t.f'tll o f .;oquations
can be solved for u(ti for arbitrary [Yi(1 +,It}, ... , Yp(t + dp )}'I' . Note that \VI ' fa. nnot solve t l ll.~ ::;ystem of equa..tions (10) fur u(l) ill c:a~t': of arbitra.ry left hand side if some rom poncnts of the vector funrt,i 0 11 .4(;c, u L as fll netiOIlS of th e control. d ~pelld fUllctiollally on the oth ers , or equiva)PIlt.iy, if rank ;~A(x. u)/iJIt < p .
for t. he analysis of discrde-time nonlinear systems. SIAM J. COlltr. and Op/m,iz., 31,1026-1044 . Halllnw r.l . (19x4) . Nonlinear sy~tcrns: sta.bility and rationality. 1nl. .f. Conir., v.40,1 - j5. Kot ta fr. (19S6). On the iuverse of discrete-tirrw lin ear-analytic system . Control- Theory and Ad-
van"cd Tcclwology, v. 2, 619- 625. Kotta, i"J. (1990). On the right inverst:' of discretc-t.imc nonlinear syste.ms. tnl . .I. Contr. , 51 : (-9. Leontaritis LT. and S.A. Billiugs (1985). Input output para.met.ric models for non-linear systems. ParI. I : deterministic lIonlilJe:u systemf';' Tnt. J. C01,t1'., vAI. :1O:1-32~. Monaco S. anti D.No rmallll-Cyro t. (1987). Minimumph a...:;e nonlirwa r disc.rete-t.ime systems and feedback stabili~ation. Pror:. 26t1r IEEE Conf. on DC8~ ,<;ion an.d Control, Los Angelfs, CA , 979 · 986. Morh a('. M. (IH90). Deterllltna.tion of inverse Vo Jt~ rr
Nijnwij f'J> H. (1989) . On dynami c dccoupling and dynamic path cont.ro llahility in economic systems. ,I OUT"lIaf of Emlwmi,: J)yna mics and Control, v . l3 ,
11 :19 . The idea [,0 gellt~rali ze the notion of right invertibility is to rcpresf'nt. the fun c.t ionally depcndr:nt. components
2411