A P P E N D I X • P R O O F S

C H A P T E R II

P r o o f o f t h e o r e m 3-5

It easily follows that f](AR)cB. Now let BeB, then according to proposition 3-3

there is a Zl_>0 such that {weB} ,~ {Wl{t,t+A]el3[[O,Zl ] for all te][}. Let (Rq) A+I

be equipped with the Euclidean inner product and let L be a matrix the rows of

which span (B[io,,a]) -L. Then {weB}-e~{wl[t,t+zaleker(L) for all te2}. This shows

6eB(AR). •

P r o o f o f p r o p o s i t i o n 8-6

In the proof we make use of two facts about polynomial matrices.

First, every submodule of Nl×q[s,s-i] is finitely generated, i.e., if

McRl×q[s,s -1] is linear and sM=M, then there exist gEN and rieRl×q[s,s-1], i=l,...,g, such that with R:=col(rl,...,rg) there holds M=M(R). This can easily

be derived e.g. from Northcott [56, proposition 1.9 and theorem 1.10].

Second, every ReRg×q[s,s -1] can be decomposed as R=UDV with UeRg×g[s,s -1] [z~ o] with A=diag(dl,...,dn) , and veRq×q[s,s -1] both unimodular and D= o o

dieR[s,s-1], i=l,...,n, where n=rank(R) and d i divides di+l, i=l , . . . ,n-1. D is

called the Smith form of R. We refer to e.g. Kailath [33, section 6.3.3].

(i) First, if BeB, then B -l- is a module, Bl-c•l×q[s,s-1], hence it is finitely

generated, so B-I-ell -1-.

Second, if B=B(R), then B-I-=M(R). This is seen as follows. If B=B(R), then M(R)cl3 "l- is evident. On the other hand, let feb J-, then we have to prove

that reM(R). Let R=UDV with U and V unimodular and D the Smittl form of R,

D=[~:], A=diag(dl,...,dn). As f e b -k, there holds {weker(R(a,a-1))}~ k v v ) N - - I N N {=eker(r(a,a-1))}. Let ~:=V~ and r:=rV =(rl,...,rq), ~ieR[s,s-1], i=l,...,q.

Then by using the fact that U and V are unimodular it follows that

{~eker(D(a,a-1))} ~ {~eker(F(a,a-1))}. For p,q~R[s,s -1] there holds

218 Appendix

{ker(p(a,a-1))cker(q(a,a-1))}.~{3cceR[s,s -1] such that q=c~p}. It hence follows

that ~i=0 for i=n+l,...,q and that there exist c~ie~[s,s -1] such that ~i=cqdi, i=l,...,n. Hence r=(cq,...,c%,0,...,0)U -l-R, and as U is unimodular reM(R).

Third, if for BeB there holds l~±=N(R), then B=G(R), which is seen as

follows. As BoB, according to theorem 3-5 there exists a polynomial matrix R*

such that 13=B(R*). It has just been shown that then B3-=M(R*), hence

M(R)=M(R*). This implies {R*(a,a -1) w=O} ~ {R(a,a -1) w=O}. Hence B(R*)=I3(R). Finally we show that f:B * B3-:B-" B j- is a bijection onto ~3_. Let .~le~3_, so

3I is finitely generated, say M=N(R), then with B:=B(R)eS there holds 133-=I~1(R), so f is surjective. Injectivity follows from the fact that if B1,B2~ , then

(ii) If dim(B-l-)=p, then there exist p elements rl,...,r p in Rl×q[s,s-1] such

that with R:=col(rl,...,rv) there holds B±=M(R). Moreover, R has full row

rank p over the polynomials. According t6 (i) B=B(R). Now suppose R also has p

rows and B=B(R). As dim(B3-)=p, £) has full row rank. According to (i)

B3_=M(R)=M(R). This implies that there exist F, ['~Rv×P[s,s -1] such that R=FR and I~=IYR. Hence (I-FYF) R=O=(I-YFF)R, and as R and ,~ have full row rank F~'=FF=I. So /~=FR with F unimodular. •

P r o o f o f p r o p o s i t i o n 3-12

D D 3_ 3- _L l Let R~A, •:=B(R). Define Vt:=vt(Lt)=[vt(13t_l+sBt_l) ] n[vt(Bz)] , t_>0. Then

clearly {vD; t_>0} forms a set of complementary spaces for ~. Let {v~0;

i=l,...,nt} be an arbi t rary basis of V D, t>_0, and d:=max{t;nt#O }. Define ( t) -1 . (~). r i =v t (v i ), i=l,...,n,, t=O,...,d, and let R be a matrix with rows r~ t),

i=l,...,nt, t=0,...,d. According to proposition 3-10 R is a tightest equation

representation of B. Moreover, in R laws of different order are evidently

orthogonM. Hence R is in (CDF) and R~R. •

Proof of proposit ion 3-13

First suppose that R is in (CDF). Clearly L+ is the leading coefficient matrix D 3_ 3_ 3_

of R, and as vt(Lt)cvt(13t)o[vt(s~t_l) ] it follows that L_ is the trailing

coefficient matrix of R. Let B:=B(R), then R is a tightest equation

representation of B. According to proposition 3-8 R is bilaterally row proper,

C h a p t e r II 2 1 9

hence L+ and L_ have full row rank. As L_ has full row rank , R (t) consis ts o f

the rows of R of o r d e r t, and as R is in (CDF) the rows of R (t) a re con ta ined D D D ±

in L t and those o f N t in Vd(LDt). As Lt_l+sLt_icI3t i t fol lows b y induct ion t ha t D 3_ ±

the rows of 17" t a r e con ta ined in Vd(S%). As va(Lt)_l_va(St_x+s6t_l) we conclude

tha t Nt±col(Vt_l,sP't_,) fo r al l t=l,...,d.

Next suppose t ha t L+ and L_ have ful l row r ank and t ha t Nt.J_col(P't_l,

sVt_l) , t=l,...,d. Then R is b i l a t e r a l l y row p rope r , hence it is a t i gh te s t

equa t ion r e p r e s e n t a t i o n of S. I t remains to show tha t laws o f o r d e r t a r e

con ta ined in L D. As L_ has full row rank , R (t) consis ts of the rows of R of

o r d e r t, and accord ing to p ropos i t i on 3-10 the number n t of rows o f R (t)

equals the dimension of vd(LDt). We now show by induct ion fo r t=O,...,d t ha t

the rows of 17" t span va(B~t) and tha t the rows of N t a r e con ta ined i n Vd(LDt).

Then the rows of R (t) a r e con ta ined in L D, as desired. ..k D

Now Vo=No consis ts of n o independent e lements in Vd(]3 0 )=Vd(LO) , hence the ±

rows span Vd(LDo). Next suppose t ha t the rows of l?t_ 1 span va(/3t_x) , then the ± .L

rows of col(Vt_l,S~Zt_l) span Vd(Bt_l+SBt_l). As Nt±col(~'t_,,s~"t_i) , the .1. ± ±

rows of Nt, which a re con ta ined in Vd(St), a re o r t h o g o n a l to va(St_l+sSt_l) ,

so t hey a r e con ta ined in va(LD). Fur ther , as L_ and L+ have full row rank N t • D

conta ins nt i ndependen t rows, and nt=dlm(Vd(Lt) ). l lence the rows of N t span ± A_ _L D

Vd(LDtt). AS 13t=13t_l+Sl3t_l+L t it fol lows t ha t ~'t:=col(P't_l, SVt_l, Nt) spans ±

Vd(S t ), which concludes the induct ion p a r t o f the p roof . •

P r o o f o f p r o p o s i t i o n 3 -14

Let B=I3(R) with R in (CDF) and the rows o rde r e d with increas ing degree . If R'

is in (CDF) with B(R')=B, then le t / / be such tha t in R"=IIR' t i le rows a re

o r d e r e d with increas ing degree . For t=O,...,d le t R (t) and R ''(t) deno te the

mat r ices cons is t ing of the rows of o rde r t in R and R" r e spec t ive ly , i.e., o f , w t - 1 . ~ t the rows t.~r=lnr)+t,...,Lr=lnr. Then vd(R (t)) and Vd(R"(t)) bo th consis t of n t

D independent e lements in va(Lt) , which has dimension nt, see p ropos i t i on 3-10.

ntxn t t tence t h e r e exis ts a nons ingula r ma t r i x Attar such t ha t vd(R"(t))=

Attva(R(t) ) and hence R"(t)=AttR (t). So R'=IIAR, A:=diag(Aoo,...,Add ).

On the o t h e r hand, if R is in (CDF) with rows o r d e r e d with increas ing

degree , then c l ea r ly AR also is in (CDF) with A=diag(Aoo,...,Aaa), Att

nonsingular , t=0 , . . . ,d , as the rows o f o r d e r t s t i l l span L D. Also lIAR is in

2 2 0 A p p e n d i x

(CI)F). As HA is invertible B(IIAR)=B(R). •

P r o o f o f p r o p o s i t i o n 3 - 1 6

P P t ± ± 2- Let ReA, S:=S(R), and define Vt:=vt(Lt)=[vt(Ft_is )+vt(Bt_1) ] n[vt(/~t) ]. We

claim that {VtP; t_>0} gives a set of complementary spaces for S. AssiLming this

to hold true, choose arbi t rary bases {v~t); i=l,...,nt} of V P, T~t):='/);l(v~t)), and let ~' have rows {r~t); i=l,...,nt, t=O,...,d}, where d:=max{t; nt¢O}. Then

according to proposition 3-10 R is a tightest equation representat ion of B,

which evidently has its rows in L B. Hence R is in (CPF) and R~/~, as desired.

To prove that {V~; t_>0} is a set of complementary spaces we have to prove P ± ± Vt+vdS t-l+SB t-D=vt( B t ). that (i) Vtnvt(Bt_l+SBt_l)={O } and (ii) P ± ± .L

P 2- 2- Concerning (i), let veVtnv t ( s t-l+sl3 t-1), say v=[ro,...,rt] , tiering,

t k 2_ .L i=O,...,t. Let r:=Zk=orkS. As ?~E]3t_l+£13t_ 1 it follows that Tt6Ft_l, and as ±

vt(r)eV P it follows that rt±Ft_,, hence rt=O. Then reSt_l , but vt(r)±vt(S¢_l) ,

hence r=O, so v=O. P .1. 2_ 2_

Concerning (ii), note that Vt+vt(13t_i+sBt_l))CVt(13t) is trivial. Now let

reBUt, then we have to show that there exist r('),r(2)~B~t_l and v ~ V P such that t k vt(r)=vt(r(1)+sr(2))+v. Let r=Zk=orkS , rt=~'t+rt, rtlFt-1, ~teFt-1. Let

2_ , t ' , k r '~t3t_l be such that r =Zk=orkS , rt,--rt,' --= where t'<t-1._ Define e(2):=

8t-t'-l.r ', then r(2)e/~t_l . Next d e f i n e ~'":=r-s'r(2)-~'t St, then d(r")<_t-1. Now , 2_

let "vt_l(r")=Vt_l(r(1))+v' where r(1)~B~_l and v±vt_,(Bt_l). Let v:=[v',Pt].

Then vtl(v)=r"-r(1)+~'tst=r-sr(2)-r(1)etg~t, as this is a linear space. t .1_ 2_ _L P

SO v ~ [ v t ( F t - 1 s )+vt(B t-l)] nvt(13 t )=Vt. Moreover, vt(r)=vt(r"+sr(2)+~tst)= vt(r(O+sr(2))+vevt(131t_l+ 2_ p sSt_~)+Vt, as desired. •

P r o o f o f p r o p o s i t i o n 3-17

First assume that R is in (CPF). Again L+ clearly is the leading coefficient P ± t ±

matrix of R, and as vt(Lt)evt(Bt)n[vt(Ft_zs )] it follows that L_ is the

trailing coefficient matrix of R. Let B:=B(R). Then R is a tightest equation

representation of /~, hence L+ and L_ have full row rank. Moreover, R (t)

consists of the rows of R of order t. As R is in (CPF), it follows that p(t ) . ,~(8) vt(R(t))±vt(Ft_lst), hence especially , - t - K s for s<t, t=0,...,d. Moreover,

it easily follows by induction that the rows of 17" t are contained i n Vd(•lt). AS

Chap te r II 221

P 2_ the rows of N t are contained in Vd(Lt).LVd(~t_I) we conclude that Nt.l.Vt_l, t=l,...,d.

Next suppose that conditions (i), (ii) and (iii) of proposit ion 3-17 are

satisfied. We have to prove that the corresponding R is in (CPF). Condition

(i) implies that R is a t ightest equation representat ion of B, see proposit ion

3-8. Hence the number n t of rows of R (t) equals the dimension of va(LP), see

proposit ion 3-10. We now show by induction for t=O,...,d that the rows of 12 t

span Vd(13&t) and that the rows of Nt are contained in vd(Let). Then the rows of

R (t) are contained in Let, as desired, and R is in (CPF). .l_ P Vo=No consists of no independent elements in Vd(So)=Vd(Lo) , hence the

P .L rows span Vd(Lo). Next suppose that the rows of Ft-I span Vd(13t_l). Condition

.L (ii) implies that Nt±vd(Ft_xSt), condition (iii) that Nt.l_Vd(Bt_t). As the rows

of Nt evidently are contained in Vd(8+) if follows that they are contained in

va(Let). Moreover, the rows of Nt are independent, due to (i), and their number i i i P

equals dim(va(LP)), hence they span vd(LPt). As 13t=13t_l+Sl3t_lTL t it follows _L

tha t Yt:=col(~'t_x,S~'t_x, Nt) spans va(Bt), which concludes the induction par t

of the proof. •

P r o o f o f p r o p o s i t i o n 3-22

In the proof we make use of a result from abst rac t realization theory, given

in the next lemma. For a proof we refer to Willems [74, sections 2.2.4, 2.4.3,

4.7.4 and 4.7.5]. For ze(Ra) z we use the notat ion z--:=z](_oo,_l], z-:=zI(_oo, o ], ++ Z Z+:=Z[[o,¢o)~ Z : = [1,00).

Lemma 3-22 Let BeB have realization Bs=I3s(A,B,C,D ) :={(v,x,w)e (Rmx Rnx Rq)Z; [ a : ] =[ADBI [:] }. Then the dimension n of the s tate space

is minimal among all realizations of B if and only if the next three

conditions are satisfied:

(1) the s tate is trim, i.e., for all x0eR n there exist (v,x,w)eB~ with

x(0)=x0;

(2) the state is past induced, i.e., {(v,x,w)e13s, w--=0} ~ {x(0)=0};

(3) the s ta te is future induced, i.e., {(v,x,w)~Bs, w+=0} ~ {x(0)=0}.

(i) Let Bd3 be given, and let Bs(A,B,C,D ) be a realization of 8 with n and m

222 Appendix

the number of state variables and driving variables respectively. Let n ' and m*

denote the smallest achievable n and m respectively. We have to show that

these minima can be achieved simultaneously.

First we derive a lower bound for ra*. For T>_0 def ine 13T(O):={a~(~)T+I;

~W~B such that w--=O, Wl[O,Tl=a } and dT:=dim(13T(O)). NOW linearity and time

invariance of B imply that dT>do(T+l). On the other hand, a realization

13s(A,B,C,D ) of /3 implies dT<_n+m(T+l ). Hence m*>d o.

Next let I3~(A,B,C,D) be a realization with n=n*. If suffices to prove

that we can reduce ra to do, as then re=m*. If [~] has column rank m'<m, then t . - - J

clearly there exists of realization of B with n=n* and m=m'. Hence we may

M J

seen as follows. Let v0eker(D), (v,x,w)EBs(A,B,C,D), w--=O, v(O)--vo, then

lemma 3-22(2) implies that w(O)=O and that O=x(O)=x(1)=Bv(O), hence vo~kcr(B). f "x

So ker(D)cker(B), which due to injectivity of I~1 implies that D is injective. % - j

By taking (v,x,w)--=O it follows from lemma 3-22(2) that im(D)=Bo(O), hence

rn=d o. So both m and n are minimal.

(ii) First suppose that (A,B,C,D) is perfectly observable, that (A B) is

surjective and that D is injective. We have to show that then n and m

both are minimal.

Concerning the minimality of n, according to the lemma it suffices to

check the conditions (1), (2) and (3) stated there. Perfect observability

inwoediately implies (3). Condition (1) is implied by surjectivity of (A B).

Indeed, let xoeR ~, then as (A B) is surjective there exists (v,x)- such that

x(t+l)=Ax(t)+Bv(t), t<_-l, with x(O)=x o. Take v ++ arbi t rary and x(t+l)=

Ax(t)+Bv(t) for t_>0. Let w:=Cx+Dv. Then (v,x,w)eB 8 and x(O)=Xo. Finally we

check condition (2). Let (v,x,w)el3~ and w--=O. Perfect observability implies

x](_0~,_n]=0. For k=O,...,n-1 one easily proves inductively that, due to

injectivity of D, there holds v(-n+k)=O, x(-n+k+l)=O. Hence x(O)=O, which

verifies (2).

Now the minimality of m follows from the proof of (i), where it was shown

that re=m* in case n=n* and IDB~ is injective, so especially when n=n* and D is M J

injective.

Next we consider the implication {n=n*,m=m*}=> {(A,B,C,D) is perfectly

observable, (A B) is surjective, D injective}.

Suppose (A B) is not surjective. Then there exists a nonsingular SeR '*×n

Chapter II 223

such that S(A B)=[A'B0']. Let Sx:=rX']LX~J be an according partition, then in

Bs(A,B,C,D ) we have x2=0. This contradic ts condit ion (1) of the lemma.

Further, re=m* and hence [B] is injective. In the p roof of (i) we have shown

tha t fo r n=n* the injectivity L~'JOf [BD] implies injectivity o f D.

Finally we have to show tha t {(v,x,w)~/3~, w[[0,n_l]=0} ~ {x(0)=0}. Now

let (v,x,w)eB s with wl[0,,_l]=0. As D is injective there exists a GcR n×q such

that B=GD. Then for O<t<n-1 we have O=Cx(t)+Dv(t) and x(t+l)=Ax(t)+Bv(t)= (A-GC)x(t)=(A-GC)txo. So C(A-GC)txo~im(D) for O<t<n-1. This implies tha t

C(A-GC)txoeim(D) fo r all t>0. Hence by choosing v I In,=) appropr ia te ly we can

const ruct (~,2,£z)eB s with (~,2,£o)[(_oo,n_ll=(V,X,W)l(_co,n_l] and w+=O. Condition (3) o f the lemma implies x(O)=O. •

P r o o f of proposition 3-25

In the p roo f we use a result on uniqueness o f the s ta te for minimal

realizations. For this result we refer to Willems [74, sections 2.4.3, 3.2.5

and 4.7.5].

Lemma 3 -25 Let /3~, i=1,2, be two minimal real izat ions of a given system

/3~B, and let n be the dimension of the s ta te space. Then there exists a

nonsingular SeR n×n such tha t fo r any we~3, if (Vi,Xi,W)~/3~, i=1,2, then

X 2 ~ - - S X 1 .

Now let Bs:=Bs(A,B,C,D ) be a minimal real izat ion o f BE[B, i.e., B={w; 3(v,x)

such tha t (v,x,w)~Ys}. First we show tha t /3s(A,B,C,D) also is a minimal rea l iza t ion of /3 if

(A,B,C,D)=(S(A+BF) S -1, SBR, (C+DF) S -I, DR) for Se~ n~n, R~R 'n×m both

nonsingular and for any F~R rn×n. I t suffices to show tha t it is a real izat ion.

This is easily seen, as on the one hand fo r web there is (v,x,w)~/3s, and with

(~.):=(R-l(v-Fx), Sx) there holds (~,w)eBs(A~B~C,D), while on the o ther hand

for (~,~,~)~Bs(A,B,C,D) there holds (v,x,w)e/3 s with (v,x):=(R~+FS-I~, S-I~),

hence weB.

Next let 13s(A,B,C,D ) be an a rb i t r a ry minimal rea l iza t ion o f /3. We then

have to const ruct S, R and F such tha t (A,B,C,D)=(S(A+BF)S -1, SBR,

(C+DF)S -1, DR). We will do this by considering i/s/o real izat ions.

224 Appendix

Let H be a permutation matrix such that in I=l :=Hw u plays the role of an LyJ Cc'~ (~1 input and y that of an output, see corollary 3-23. Let IIC:=t lj and HI):= 12 ,

where the partitions are according to ( : ] . T h e n 13s(A,B ,[cc: l:ci~:]) is a minimal

realization of IIB. Now D 1 is invertible, which is seen as follows. Let m u

denote the number of components in the (free) variable u. Let 13o(O):={aeRq; 3[~IEIIB with [~1--=0, [~l(0)=a}, then according to the proof of part ( i ) o f

proposition 3-22 there holds dim(13o(O)):=do=m*, while clearly do>_mu. The fact

that 6eB has i/s/o realizations implies m*<mu, hence do=m ~. Moreover, due to

minimality the state is past induced, from which it follows that g0(0)=im[D:].

As u is free it follows that DleR m*Xm* is surjective, hence invertible.

Defining {A,B,C,D):=(A-BD-11C1, BD-11, C2-D2D~IC1, DzD-11) we get 1113={ [~] ; 3(v,x)

such that [a~]=[AIB1]LC2 D2J [:]}={[~); 3X such that [ a ~ ] = I ~ l [:1}.

For (A,B,C,D)we analogously get //13={[~1; ~£ such that [ a ~ = [ ~ - ] [~}

where (~4,B,~f)):=(~4-Bt)-1'CI, / ~ 1 , ~2_~2/~1~,, ~2~[1). As these are two

realizations of the same system, according to the lemma there is an S such

that £=Sx. Hence (x ,u ,y ) s a t i s f i e s [~1 = [~ ~1 [ : ) i f and only if

[ ~ ] = [ s ~ s-l~] [:]. As the state is trim and past induced we can generate

anyxoeRnandtakeu(O):O, fromwhichweconclude[S-cTtss]:[~].Bytakingxo=O

and u arbitrary we conclude I s - l ~ [ ' ] A direct calculation now shows that t B J = LDJ (A,B,C,D)=(S(A+BF) S -~, SBR, (C+DF) S -1, DR), where R:=D~I/~I is invertible

and F:=D-11(C1S-C1). •

Proof of proposi t ion 3 - 3 2

In the proof we use a result on representation of linear, time invariant,

complete systems for time axis T=N. A system B¢(Rq) • is called linear if it is

a linear subspace of (Rq)t~, time invariant if B li2,00)cB , and complete if

{weB} *~ {wl[1,tleB][1,t] for all teN}. By B~ we denote the class of linear,

time invariant, complete systems in (Rq) ~. By B~(AR) we denote AR-systems in

(Rq) ~, i.e., any set for which there exist geN and ReRg×q[s] such that

B=B~(R):={we(Rq)~; [R(a)w](t)=O for all teN}. The following result is analogous

Chapter II 225

to theorem 3-5 and follows from Willems [73, theorem 5].

Lemma 3-32 B~=B~(AR). Moreover, for any BeB® there exists a row proper R

such that B=I3®(R).

Now let BoB T and define Be:=[we(Rq)~; W I[t,t+T_I]E]~ for all teN}. Shift

invariance of B implies 13=Bel[1,T]. Moreover, BeeB~. Let ReRg×q[s] be row

proper such tha t Be=lJoo(R). We will show that for any row proper R there holds

Bt(R)=B~(R)[[1,t], for all teN. Then especially 13=Boo(R)I[1,T]=I3T(R), which

shows BeBT(AR ). • , ~ a i ( i ) t~ (ii)~=0, i e [ 1 , g ] . Let RcRgXq[s] be row proper with rows ri(s)=2~lc=or k s , r d

Let L+:=col(ra(ii); ic[1,g])eR g×q. Then L+ is surjective as R is row proper.

Now first let wel3~(R)[[1,t], say w=Col[1,t] with ~eB~(R). Then

[ri(a)~o](T)=O, r~N, and especially for rows of R with degree di<t-1 there

holds [ri(~r)w](7")=O , ~-e[1, t-di] , which by definition means that wcl3t(R ).

Conversely, let weBt(R ). Then define ~e(Rq) t~ recursively as follows. Let

~[[1,tl:=w. If ~ is defined on [1,t*], then define ~(t*+l) as an arbitrary

solution of r{dl)~o(t*+l)+r(j.)_l~o(t*)+...+r(oi)~(t*-di+l)=O for all i with di<_t*. Existence of a solution is guaranteed as L+ is surjective. Clearly ~eB~(R), as

all laws [R(a)~z](t)=O, teN, are satisfied. Hence wel3~(R)[[1,t I. •

P r o o f o f p r opos i t i on &-33

(i) First let BoB T. It follows from the proof of proposition 3-32 that there

exists a row proper R with d(R}_<T-1 such that 13=I3T(R ).

Conversely, let d(R)<T-1 and R row proper. Let R have rows ri(s)= Z~t i ( i )k r(1)~0 ' ic[1,g], and let L+ee g~q have rows r (i) ie[1,g]. To show k=OT"k 3 , di ,

that BT(R)eBT it suffices to consider shift invariance, i.e.,

aBT(R)CBT(R)[D,T_I]. Now this condition is equivalent to existence of a

solution ae~ q of the set of equations r(a:)a+r(cl:)_,w(T)+...+r(ol)w(T-di+l)=O for

all ie[1,9], for fixed weBT(R ). Because R is row proper, L+ is surjective and

existence of a solution is guaranteed.

(ii) If R is zero order bilaterally row proper, d(r)_<T-1, then one easily

shows by a similar reasoning as in (i) that BT(R)eBr.

Conversely, let BeB T. Define ~ee:={'//}E(Rq)Z; W[[t,t+T_l]~13 for all teT}. As

226 Appendix

S is t ranslat ion invariant it follows that B=Bee[[1,T]. Moreover, 19eeeB.

According to theorem 3-5 and proposition 3-8 there exists a bilaterally row

proper R such that Bee=B(R)e8. I t follows from proposit ion 3-10 that R can be

chosen to be zero order bilaterally row proper by multiplying the rows by

appropr ia te f ac to r s a hi, nieT/. It remains to show tha t ~(R)_<T-1 and that

13(R)][,,T]=BT(R ). This follows by a reasoning completely analogous to the one

given in the proof of proposition 3-32, using the fact that R is bilaterally

row proper. •

P r o o f o f l emma 3 -34

In the proof we use a result for systems in Be0 which is analogous to

proposit ion 3-6(i). The proof is completely analogous to that of proposition

3-6. We use the following notation. For BeBm let B±:={reRl×q[s]; [r(a)w](t)=O, for all web and all teN}. If R~Rg×q[s] has rows rieRl×q[s], then let

M~(R):={re~l×q[s]; 3pieR[s], ie[1,g], such that r=zg=lpivi) denote the

submodule of Rl×q[s], generated by the rows of R.

Lemma 3-34-1 Let BeS~, Then {6=/3~(R)} ~=> {B±=Moo(R)}.

Now let BT(Ri)eBr, i=1,2. Define B~:={we(Rq)~; W][t,t+T_l]el3T(Ri) for all teN},

i=1,2. Shift invariance of BT(Ri) implies that B~I[1,T]=I3T(RI) , i=1,2, and ¢ ¢

that {BT(R1)CBT(R2) } <~. {BlcB2}.

Define Boo(Ri):={we(Rq)~; [Ri(a)w](t)=O for all teN}, /=1,2. Because [l(Ri)< e

T-1 there holds Bi=B~(Ri), i=1,2, which is seen as follows. Let R i have rows

r~.i),je[1,gi]. If W~Boo(Ri) , then [r~.O(a)w](t)=O for all je[1,gi] for all

teN, especially for all te[v, ~'+T-d(r}O)-l], for &ll teN, so w[[~,r+r_lle

BT(Ri) for all yeN, hence weB~. Conversely, if weB, e. then with wt:=w[[t,t+T_l] , teN, wteBT(Ri) , so [r}i)(a)wt](r)=O for all je[1,gi], for all re[1,T- d(r}0)]¢O, especially [r}O(a)wt](1)=[r}i)(a)w](t)=O for all teN, hence

weB~(Ri).

So to prove lemma 3-34 it remains to prove that {19~(RI)CI3~(R~) } ~=~ {there

exists an F such that R2=FRI}. Now (~) is obvious, while (=>) follows from

lemma 3-34-i. •

Chapter II 227

Proof of lemma 3-35 and proposi t ion 3-36

In the proof we make use of the following result, which was shown in the proof

of proposition 3-32.

Lemma 3-35-1 If R is row proper, then for all TeN ]3eo(R) I[1,TI=I3T(R ).

First we consider the results for LCLM and addition.

Let Ri be row proper, i=1,2. Define 13co(Ri):={w~(Rq)N; [Ri((7)w](t)=O for

all teN} and B:=Yoo(R1)+Boo(R2). It follows from lemma 3-32 in the proof of

proposition 3-32 that there exists a row proper R 0 such that B=13¢0(R0). We will

show that RocLCLM(RI,R2). Row properness implies that for all TeN 13T(Ri)=13(z(Ri) ]D,T], i=1,2, and

that 13T(Ro)---- Boo(Ro)[[1,T]. As [Boo(R1)+/~co(R2)][[1,T]= B~(R1)][1,T]+Bco(R2)][1,T] this implies that for all TeN also BT(R1)+BT(R2)=BT(Ro).

Taking T>_max{d(R0), d.(R1) , d(R2)}+l, l~T(Ri)CI3T(Ro) by lemma 3-34 implies

that there exist F~ such that Ro=FiR i. Moreover, if for /~ there exist /~i with

R=FiR,, i=1,2, then let U unimodular be such that UR=I~ ] with k r o w proper,

cf. e.g. Wolovich [77, theorem 2.5.11]. Then Yoo(R)=Sco(R)DSoo(R1)+13®(R2)= , u j

B®(Ro). Due to row properness it follows that for all TeN BT(R)DBT(Ro). Taking

T>max{~l(Ro) , d(R)}+l it follows from lemma 3-34 that there exists an F such

that [~=FRo, hence R=FR o where F:=U-lfFo]. This proves that RoeLCLM(R,,R2)and x. j

(i) of lemma 3-35 for LCLM.

Next let ReLCLM(R1,R2) be row proper. Then there exists an F 0 such that

R=FoR o and an F such that Ro=FR. In the notation of lemma 3-34-1 hence

Mo~(R)=Moo(Ro) , so Bco(R)=B~o(Ro)=Bco(RI)+Bo~(R2). Due to row properness BT(R)= I~T(R1)+BT(R2) , which proves (~) of proposition 3-36. Moreover, R=FoFR and

Ro=FFoR o. Because R 0 and R are row proper they have full row rank over R[s],

so that FoF=FFo=I, which proves (ii) of lemma 3-35 for LCLM.

Now second we consider the results for GCRD and intersection.

Let R~ and R~ be row proper. From lemma 3-32 in the proof of proposition

3-32 it follows that Boo is closed under intersection, as Boo(AR) clearly has

this property. Moreover, according to this lemma there exists a row proper R 0

such that B®(Ro)=B~(R1)nt3o~(R2). That RocGCRD(RI,R2) is proved in a way

228 Appendix

analogous to the result for LCLM, and one easily gets lemma 3-35(i) and (ii)

for GCRD.

To prove proposition 3-36(ii), let RcGCRD(R1,R2) be row proper and let

d.(Ri)___T-1 , i=1,2. By lemma 3-35(ii) there exists a unimodular Y such that

R=URo, so Boo(R)=B®(Ro) and BT(R)=BT(RO). So it suffices to prove that

BT(R0)=E{~T(R); ]3T( R, )CI3T( R1)NI3T( R2) }. Now Boo(Ro)ct3o~(Ri) and row properness implies 13T(Ro)CST(Ri) , i=1,2; hence

BT(Ro)Cl3T(R1)nBT(R2). So it suffices to prove that for R row proper

{13T(R)Cl3T(R1)~I3T(R2)) =¢, {13T(R)Cl3T(RO) }. Let t2 consist of the rows of R of

degree at most T - l , then d(Ri)<T-1, /=1,2, and lemma 3-34 implies that there

exist Fi such that Ri=FiR, i=1,2; hence B®(l~)cB®(Ro) and BT(~))=BT(/2)C

GT(R0). •

C H A P T E R III

P r o o f o f p r o p o s i t i o n 3 -7

Let P be bilaterally monotone on 13~ T. By taking t=2 in the definition of

bilateral monotonicity we have that generically in we6 {Gr_l(R)e

PT_I(WI[2,T]), ]3'ePTW}=~ {aB'eBT_I(R)} =~. {B'CBT(aR)} , which proves the shift

invariance condition for t=T.

To prove this condition for general te[2,T], let Be_I(R)ePe_I(wI[=,t] )

and B'ePt(wl[l,t]). Now Be~r , so by proposition II.3-30(ii) there exists ~eB

with "W[[T_t+l,Tl=Wl[1,t] and ]3t_I(R)ePt_I('W[[T_t+2,T]), B'EPt('W[[T_t+I,TI ).

NOW bilateral monotonicity implies that generically in CoeB al3'cSt_l(R), hence

B'eBt(aR ). We have to prove that this holds generically in weB. It is

sufficient to construct a linear bijection w-~ ~. This can be done as follows.

Let Bee:={wee~(Rq)Z; wee][r,r+T_lleB for all re][}. Because B is e e

t ranslation invariant B I[1,TI=B. I t can be shown that there exists a linear

injection L:8 -> B~e:w -> w ee w i t h ~#ee][1,Tl~-W such that for all f e z Lr:B -> G:

w -> Wee[[~+l,r+T] iS a bijection. Then for ~zeB take ~:=Lt_TW.

The idea to construct L is as follows. Let 6i/s/o be a minimal (forward)

input /s ta te /output realization of 8 ee (see corollary II.3-23) with state x and

with / : / = / / w a corresponding input/output decomposition of w. I t can be shown , . - .%

that there exists a linear map f such that x(r)=f(w[[r_r,r_d) for all reP. e e e e

Now take in 13i/8/o u periodic on Z+ with u I[kT+I,(I,.+I)T]:=UI[1,T] ~ k~.N.

Together with x(T+l) this uniquely defines "weel[o,~]. We define wee](_®,_1] in

an analogous way, using a backward realization 8i/s/o of Bee. This defines a

linear injection L:B--> B ee To see that Lr is a bijection, suppose that

wee][r+l,r+T] is given for some feZ. From this we can reconstruct x(r+T+l) and

u, as u is periodic. From Bi/s/o and 8i/s/o we then can reconstruct W ee o n ~'~

W e e hence especially :=w [1,T]. •

230 Appendix

P r o o f o f lemma 3-11

(~) Let r(n) be linearly independent from r(1),. . . ,r(n-1) and r(n+l) linearly }2

dependent on r(1),...,r(n), say r(n+l)=$i=lair(~) (defined for the columns

1,...,T-n of HT(W)). Define w(r), r>T, recursively by w(T):=En=laiw(T-n-l+i)

and define a Hankel extension M of liT(w ) by mij:=w(i+j-1). Using the ttankel

structure one gets rank(M)=n, hence rank(HT(W ))<_n. To prove that

rank(HT(W))>n , let M" be an arbitrary extension of IIT(W) and let d:=rank(M').

If d<n this would imply that among the rows 1,...,n of M' at least one, say

row n', is either zero (in which case r(n') is linearly dependent on

r(1),...,r(n-1) by definition) or it is linearly dependent on the foregoing

ones. This implies that r(n') is linearly dependent on r(1) , . . . , r (n ' - l ) , and

because of the Hankel structure of HT(W) and the fact n'<n this means that

r(n) would be linearly dependent on r(1),...,r(n-1). So rank(M')>n and hence

rank(HT(W))_>n.

(=,) Let rank(HT(W))=n. Then r(n) cannot be linearly dependent on

r(1),. . . ,r(n-1), as the construction above would give rank(HT(W))<_n-1.

Moreover, r (n+l) cannot be linearly independent from r(1),...,r(n), as this

would finply that any extension of liT(w ) would have rank at least n+l. •

P r o o f o f lemma 3-12

A minimal rank extension which is Hankel was constructed in the proof of (~)

of lemma 3-11. •

P r o o f o f p r o p o s i t i o n 3-13

Let rank(HT(W))=d. Assume w~B~B T. Shift invariance of B implies that there

exists weE(R) •, We l[1,Tl=W, We[tr,r+r_lleB for all v~N. Define an extension /~I

of liT(w ) by mij:=we(i+j-1), i,j=l,...,T, mij:=O elsewhere. Then rank(M)>d and

hence we l[r,r+T_l], T~[1,T], span a space of dimension at least d in B, hence

c(t3)>__d. Further, as rank HT(W)=d there exists a=(al,..,aa)eR d such that d . d d i-1 r(d+l)=~i=lair(z). Define Ra:=a -Zi=laia Then clearly weI3T(Ra) and

c(13T(Ra))=d. Using the definition of PT g this proves (i) and 3 in (ii). To

prove c in (ii), let BT(R)ePKTTW, so C(BT(R))=d , which implies that R has degree

Chapter III 231

d ~ d i-1 d . d, say K=a -2.~=laia . Then in HT(W ) r(d+l)=Y~i=lair(z ). •

P r o o f o f proposition 3-16

Let R~0 have degree [I(R)=d. First assume d<ENT(T/2), so we have to show tha t

generical ly in welt(R) rank(HT(W) )=d. For wel3T(R ) row d+l of HT{W ) is l inearly dependent on the foregoing ones,

so rank(HT(W))<_d. To prove tha t gen. rank(HT(W))=d, it suffices according to

lemma 3-11 to show tha t gen. row d is l inearly independent f rom the foregoing

ones. Sufficient fo r this is tha t gen. in WeBT(R ) rank(Hd, d(w))=d , where

w(1)

Hd,d(~):= ~ ( 2 )

v(d)

~ ( 2 ) ... w(d )

w ( 3 ) ... w ( d + i )

w ( d + l ) ... w ( 2 d - 1 )

Note tha t d<_ENT(T/2), so 2d-l<_T and Hd, d(i/J) is well-defined.

In BT(R), w(r), Te[1,d], can be chosen arbi t rar i ly while w(r), r e [d+ l ,T ]

can be expressed as linear functions of w(r), ve[1,d]. So for WeBT(R ) det(Hd,~(w)) can be considered as a polynomial in (w(1),...,w(d))e~ d. It

suffices to show tha t det(Hd,d(W)) is not the zero polynomial , because then

{w;rank(Hd,d(w))<d}={w;det(Hd, d(w))=O} is a proper algebraic va r i e ty and

hence rank(Ha, d(W))=d gen. in W~6T(R ).

That det(Hd, d(w))$O is seen as follows. We claim tha t det(Hd, d(W)) contains {w(d)} d as a term with coeff icient _+1. Indeed, det(Hd, d(W))=

. d p~p{Slgn(p).lli=la~p(i)} , where Hd,d(W)=(aij), P denotes the set o f all

permutat ions o f {1,...,d}, and s ign(p)e{-1 ,+l} . In order to get {w(d)} d, f r o m

every row and column i n Hd,d(W ) one has to choose an element which involves

w(d). In the f irst row this is only the element (1,d), so p(1)=d. In the

second row only the elements (2 ,d- l ) and possibly (2,d) contain w(d), so

necessari ly p(2)=d-1. Going on in this way one gets f o r {W(d)} d the unique

permuta t ion p:={d,d-1,...,2,1}. This proves our claim and hence det(Ild, d(W))~tO. Next assume R=O or d>ENT(T/2). By a shnilar reasoning as before one can

show tha t gen. in weBt(R ) HENT(T/2),ENT(T/2)(w ) has rank ENT(T/2) and hence

row ENT(T/2)+I o f HT(W } then is l inearly dependent on the fo regoing ones (its

length is ENT(T/2) if T is even, ENT(T/2) - I if T is odd). So then

rank(HT(W))<ENT(T/2 ) and hence it equals ENT(T/2), as row ENT(T/2) is l inearly

232 Appendix

independent from the foregoing ones. •

P r o o f o f t h e o r e m 3-17

(i) Obvious.

(ii) Not monotone. Let 13=R T and te[3,T] odd. Then according to proposition

3-16 gen. in web there holds that for Bt_ld~t_1(w[[1,t] ) and Bt~P~t(w[[1,t]) dim(Bt_l)=(t-1)/2 and dim(6t)=(t+l)/2. For t>3 (t+l)/2<t-1 and dim(Bt)=(t+l)/2 implies that dim(Bt[[1,t_ll)=(t+l)/2 so gen.

Bt[[1,t_l]¢Bt_ 1 and pK is not monotone. We have used the fact that {GeEt,

dim(/~)=d} =~ {dim(13[[1,r])=d for all ve[d,t]} which follows from {6eEt,

dim(B)=d} ~, {there exists R of degree d such that 13=13t(R)}. Not shift invariant. Let B=R r and take t=3, so B l[2,t]=R 2 and B[[1,t]=R 3.

Let weB, wi[1,3]=(a,b,c ) with a¢O, b~O, ac-b2~O. Then P~2(w[[2,3])=132(¢-(c/b)) and Ba(a2-(c/a))eP~'3(w[[1,3] ). Shift invariance would require that gen.

I33(a2-(c/a))cBa(a(a-(c/b)))=B3(a2-(c/b)a), which clearly does not hold true.

Not linear. Take T=3, 131:=B3(a2-1),B2:=B3(~r+2). Then Bl+B2=R 3 so

generically in (wl,w2)eBlX/~2, if BeP~(wl+w2) then dim(B)=2. Also generically

BleP~3wl and 62=P~3w 2. Linearity would require that Ra=BI+B2cB which is false.

(iii) Take for example T=3, B0=R 3. Then gen. in weBo, if 13ePgw then dim(B)=2,

so B0¢B.

(iv) We will determine im(P~), ~KT, B~TK.

That im(P/~T)=B:r is seen as follows. If B=~ T then take weff~ T defined by

w(t)=0, t~[1,T-1], w(T)=l, so rank(HT(W))=T and by proposition 3-i3(i) RT=p~TW. If RT~BEBT, then according to proposition IL3-33(i) there exists R with

d=[l(R)<T-1 such that B=BT(R). Choose wel3T(R ) with w(r)=0, T6[1,d-l], w(d)=l and w(r) for re[d+l,T] computed by means of R. Then rank(HT(W))=d and by

proposition 3-13(i) BT(R)ePZT w. Next we prove B~={BeET; c(B)<_ENT(T/2)}. From example 1 we know already

' that R T ~ . Now let RT~BeET and let R¢0 with c(B)=d=~(R)<T-1 be such that

B=BT(R ). If d>ENT(T/2) then from proposition 3-16, gen. in weI3T(R), if 6"ePTw then dim(B)=ENT(T/2), hence I3T(R)~t@TW , So BT(R)~B~. If d<ENT(T/2) then gen.

in WeBT(R ) rank(HT(W))=d , so BT.(R)eP~w and BT(R)eEU~K. Finally we consider B ~ . First let T be even. Let BeE T with d:=c(B)<

ENT(T/2). Then gem in web the first d rows of IIT(W } are linearly independent

C h a p t e r III 233

and row d+l has T-d>_T/2>_d elements and this row is linearly dependent on the

foregoing ones. Using proposition 3-13(ii) and the linear independence of rows

1,...,d, this implies that gen. in web P~TW={B), i.e., the assigned model is

unique. So B ~ B ~ , hence equality holds.

Next let T be odd. If c(B)<_ENT(T/2)-I then by a reasoning as before we

get B~B~. If d:=c(B)=ENT(T/2) tllen in HT(W ) row d+l consists of T-d= (T-1)/2<d=(T+l)/2 elements and gen. pKTW is not unique; hence B~B~TK.

(v) As can be seen from the reasoning in (ii), (iii) and (iv), lacking the

properties of monotonicity, linearity, truthfulness and prudence has not

to do with possible nonunique assignment of models by pK. We now show that

shift invariance also cannot be obtained by choice of a selection rule S.

To get shift invariance, taking the example in (ii) with a~O,b~O,c#O, K ac-b2~O, this would require that for BeP3(a,b,c ) BcB3(a~-(c/b)a) while

dim(B)=2, so this requires Sa(a,b,c)=Ba(a2-(c/b)a). Moreover it is required

that (gem) S4(d,a,b,c)cB4(aa-(c/b)a2). Now gen. if BePK44(d,a,b,c) then

dim(B)=2. Let B4(a2+c~ct+fl)~(d,a,b,c). In order that B4(a2+c~a+fl)cB4(a a-

(c/b)a 2) according to lermna II.3-34 there has to exist a T such that

(a2+(xa+fl)(a+T)=a3-(c/b)a 2, which implies that (c~,fl)--(0,0) or (c~,fl)=(-c/b,O). But B4(a2)~t~4(d,a,b,c), as it requires b=c=0, and B4(a2-(c/b)a)~l~4(d,a,b,c), as it requires ac-b2=O. So it follows that it is impossible to construct a

shift invariant selection rule for pK. •

P r o o f o f t h e o r e m 3 - 2 0

The proof of this theorem is quite lengthy and will be split in a number of

steps. The result is proved by using a number of lemmas, some of which play a

role in the proof of proposition 3-25.

Nota t ion. First we introduce some notation. T is assumed to be fixed T - I k throughout. For R=Zk=oaks eR[s] let I(R):={ke[O,T-1]; ak=0}, so #I(R)=T-c(R).

Let B*(d), B*(I) and B*(d,I) as subsets of B T be defined as follows.

B*(d):={BT(R)eB~; ~(R)=d}, B*(I):={BT(R)eBT; I(R)~Z}, B*(d,I):=B*(d)nB*(I). Moreover define W(d), IV(I) and W(d,I) as subsets of R T by W(d):=U{BT(R); BT(R)~B*(d)}={wE~T; 3BT(R)~B*(d ) with w~BT(R)} , W(I):=U{BT(R); BT(R)~B*(I)} , and W(d,I):=U{BT(R); BT(R)~3*(d,I)}. Without loss of generality we restrict

2 3 '1 A p p e n d i x

attention to (d,I) with d+(T-#(I))<_T, i.e., d<_#(I). []

Let B0e/~ T be fixed, web o and HT(W ) its incomplete Hankel array. We now first

give an outline of the proof of theorem 3-20 by means of four lemmas and then

will give the proof of these lemmas.

Lemma 3 - 2 0 - 1 For every (d,I) either (i) w~W(d,l) generically in weBo, or

(ii) BoclV(d,I).

Lemma 3 - 2 0 - 2

19T(R(d,I) ) }.

{BocW(d,I)} ~ { 3BT(R(d,I))~B*(d,I ) such that B0c

This lemma states that if for every weBo there exists a model 13w(R)cB*(d,I )

such that weBw(R), then there exists such a model independent from weBo.

For Bo define d0e[0,T ] as follows. If EoCW(d,I } for all I and de[0,T-1]

then do:=T , else do:=min{de[O,T-1]; 3I such that BocW(d,I)}.

If d0=T , then according to lemma 3-20-1 (and using the fact that the

number of indexsets I is finite) generically in web o w¢W(d) for all de[0,T-1].

This means that gen. row d+l of H T is not linearly dependent on less than T-d

foregoing rows of IfTI so P~w=R T gen. on B0, and theorem 3-20 follows as

obviously in this case there is no BT(R)eB ~ with BoCBT(R ) and C(BT(R))<T.

For doe[0,T-1] let J0 be defined by Jo:={I;BocW(dD,I)}. Because by lemma

3-20-1 gen. on B 0 w¢IV(d) for d<d o and by lemma 3-20-2 BoCBT(R(do,I)) for

t~Jo, for some R(do,I), it follows from the definition of P~ that gen. on B0 0 * {BT(R(do,I)); Iedo}CPTWcB (do). Indeed, on B0 gem no remarkable laws of

degree d<do are satisfied while remarkable laws of degree d o always exist.

Because of lemma 3-20-1 we even have that gen. on B 0 0 * { BT( R( do,I) )'~ [~.Io}CPTWcU{B (d0,I); IeJo}.

0 * Lemma 3 - 2 0 - 3 For IeJo gen. on Bo PTWnB (d0,I) is a singleton, i.e.,

BT(R(do,I) ).

The generic way in which P~ assigns models on the basis of data from B 0 is

described in lemma 3-20-4, which is a direct consequence of lemma 3-20-3 and

the preceding discussion.

Chapter III 235

Lemma 3 - 2 0 - 4 Generically for wel3 o P~-w={l~T(R(do,I) ); IeJo}.

Now from lemm& 3-20-2 and lemma 3-20-4 it follows that gen. on B 0 if BePTW

then B0c• and c(B)=d o. To conclude the proof of theorem 3-20 note that by

definition of do if BeB r with c(B)<do, then B0¢B.. On the other hand, if 0 BeBT, c(B)=do, BocB , then gen. B~PTW. This proves theorem 3-20.

Finally we prove the foregoing lemmas.

P r o o f o f l emma 3-20-1 and lemma 3 - 2 0 - 2

Let (d,I) be given, c:=T-#(I), and let c+d<T. Suppose that wf~W(d,I) is not gem

true on B0. We will show that then there exists RcR[s] with 13T(R)c•*(d,I ) and

BOCBT(R), which proves the desired results. The proof has the following

structure. First we introduce some notation. Next in step (i) we prove the

result under two conditions, (C1) and (C2) , and using an auxiliary lemma, (L).

In step (ii) we prove (L) under conditions (C1) and (C2). Finally in step

(iii) we consider the case where condition (C1) or (C2) is not satisfied.

N o t a t i o n . Let [O,T-1]\I={il,ie,...,ic_l,d } c+d<_T. For web o define Hz(w)cR cx(T-d) by

with 0<i1<i2<...<ic_1<d and

HI(w):=

w(i 1+1) w(i 1+2) ... w(il+T-d )

w(i 2+1) w(i2+2) ... w(i2+T-d)

w ( i c _ l + l ) w(ic_x+2) ... w(ic_[+T-d )

w(d+l) w(d+2) ... w(T)

Let BO=BT(Ro) with Ra=ro+rls+...+rn_lSn-l+s n, where n:=dim(/30). Note that

wcW(d,I) if and only if the last row of Hz(w ) is l inearly dependent on the

foregoing ones. As it is given that this is not gen. false on B0 it follows

that Bo¢R T and hence n<_T-1. Let el:=(1,0,...,0)eR 1×" and define AeR n×" by the

so-cal led companion matrix of R0, i.e.,

236 Appendix

A:= 1 ° ° ' 1°1 0 1 0 . . . 0

0 0 . . . 1 0 = 0 0 . . . 0

L-To - r l - r 2 . . . - r n - 2 - r n - 1

0 In-1 ] .

I-to (-rl...-r._~)j

For we13 o le t x:=col(w(1) , . . . ,w(n))eR n. Note tha t {we13o}~=~{3xeR n such tha t

w(t)=elAt- lx fo r all te[1,T]}, so (A,el) is a "minimal rea l iza t ion" of 130, cf.

def ini t ion I I .3-20 and propos i t ion II.3-22(~i). Finally fo r xeR n define

x(t):=elAtX, t~yz+, and define H(x ,k )eR c×k by

H(x,k):=

x(il) x(G)

X ( i l + l ) . . . x ( i l + k - 1 ) x ( i 2 + l ) . . . x ( i 2 + k - 1 )

x(ic_x) x ( i c _ l + l ) . . . x ( i c _ l + k - 1 x(d) x ( d + l ) . . . x ( d + k - 1 )

Let Hi(x ) :=H(x ,T -d ) and II~(x):=H(x,2n-1). Note tha t f I i (x)=lI~(w ) for

w(t):=elAtqx, te[1,T]. []

(i) As a f i rs t s tep we p rove the result under condit ions (C1) and (Ca) and

using the aux i l i a ry lemma (L).

(C1) B0=BT(R0) , Ro=ro+rlS+...+rn_lsn-l+s n, where r0#0;

(C2) gen. on Y 0 the ma t r ix consist ing of the f i rs t c-1 rows of Hi(w) has

full row r ank c - l ;

(L) if w¢IV(d,I) is not gen. t rue on 13o, then under condit ions (C~) and

(C2) there holds tha t fo r generic xeR" the last row of H~(x) is

l inear ly dependent on the fo rego ing ones.

So suppose (L) holds true. Note tha t fo r Xo:--(O,...,O,1) T the re holds

det([xoAxo.. .An-lxo])#O, hence fo r gener ic xeN n de t ( [xAx . . .Anqx])¢O. This

implies tha t there exists xcR n such tha t this condi t ion is sat isf ied and

m o r e o v e r the last row of tt~(x) is l inear ly dependent on the foregoing r~C-1 jik+t

ones, i.e., there exist vckeR , ke[1,c-1] , such t ha t elAd+tx = 2,k=lc%elA X

fo r all t e [0, 2 n - 2]. Then col{ el, e lA, . . . , e lA "-1 )Ae[ x / i x . . . An-ix ] = _ n - I v-,c-I . i k , r . . A n - I x col(ex, e l A , . . . , e l n )(~k=lc~kn Jt X Ax. ], and as bo th col(%exA,

n-1 ,d r~c-1 Ai k ...,elA } and [xAx . . .Anqx] are inver t ib le it follows tha t zl =~k=lC~k/l . n d ~ c - 1 i k Define K:=s-2,k=lcYkS , then c lear ly BT(R)eB*(d, I ) and B0cBT(R), as fo r w~13 o

Chapter III 237

_ t - l , _ d ~ c - 1 - - i k . ^ and x:=col(w(1),...,w(n)) there holds [R(a)w](t)= elA (A-:.k=ltxk.q )X----O,

te[1,T-d]. This proves the desired results.

(ii) As a second step we prove (L) under conditions (C~) and (C2). If w¢W(d,I)

is not gen. true on Bo, then rank(Hl(X))=c is not gen. true on R n, and as

this is a polynomial condition in x we conclude that rank(Hl(X))<_c-1 for all

xeR n. We state that gen. the first c-1 columns of HI(x) are linearly

independent. Suppose this would not hold true. Then those first c-1 columns

are always linearly dependent and there is a column k<c-1 which gen. is

linearly dependent on the foregoing ones, say for xeV~{R'~\p-l(o)} for some n ~ ~ T - d - k

polynomial p:R ->R with p#0. Let pi:=pA i, ie[0,T-d-k] , and p:=tli=o Pl, then ~#0

as A is invertible under condition (CI), and V:={x; AixeV for all

ie[O,T-d-k]}~Rn\~-l(O) is generic in •n. For xeV there holds Ax~V and from the

structure of Iti(x ) it follows that for such x column k+l of Hi(x) is linearly

dependent on columns 2,...,k, hence also on the first k-1 columns. Analogously

it follows by induction that for xeV all columns k+i of Hi(x ) are linearly

dependent on the first k-1 columns, i~[O,T-g-k], and hence gen.

rank(Hl(X))<k-l<c-2. This contradicts condition (C2).

So gen. the first c-1 columns of Hz(x ) are linearly independent and as

rank(Hz(x))<c-1 for all xeR n it follows that gen. column c of H~(x) is

linearly dependent on the foregoing ones, say on the generic set V'cR n. Then

V':={x; AIx~V ' for all O<_i<_max{2n-l-c, T-d-c}} is also generic, due to (Cl) ,

and for xeV' both rank(Hz(x))_<c-1 and rank(H~(x))<_c-1. If T-d>_2n-1, then (C2)

implies that gen. on A n the last row of Hi(w)=Hi(x ) is linearly dependent on

the foregoing ones, and hence the same holds true for tl}(x) as it contains

less columns than Hi(x ). If on the other hand T-d<2n-1, then (C2) implies that

gen. the first c-1 rows of H~(x) have rank c-l, and hence again gen. the last

row of Hei(x) is linearly dependent on the foregoing ones. This proves (L).

(iii) Finally we consider the case where condition (C1) or (C2) is not

satisfied. This step is split in four parts.

(iii-1) First suppose that condition (C2) is not satisfied. Then there exists

a k_<c-1 such that the matrix consisting of the first c-1 rows of HI(w)

always on B o has rank at most k and such that there is a WoeB 0 with

rank(H1(wo))=k, say for Wo the rows in J':={i~,...,ik}c{il,...,ic_1}=:J are

238 Appendix

linearly independent, where O<_i~<...<i~<d. Then gen. on B0 the rows in J ' are

linearly independent and the rows in J\J' are linearly dependent on those in

J ' . As it is supposed that the last row of Hl(W ) is not gen. linearly

independent from the rows in J it also is not gen. linearly independent from

the rows in J ' . Now condition (C2) is satisfied for c ' :=k+l and

I':=[ O,T-1]\ ( J'u{ d} ).

I f condition (C1) is satisfied then the results of (i) and (ii) imply

that there is an R' such that 13oCl3r(R')cB*(d,I')d3*(d,I ) as 1%1, and the

desired results follow.

Finally let condition (C2) be satisfied for c' and I ' as defined before

and suppose that condition (C1) is not satisfied. Let n':=min{k; rk¢0 }. We will

consider three cases, i.e., in (iii-2) that n'>d, in (iii-3) that n'<i[, and

in (iii-4) that i~<n'<_d.

(iii-2) The case n'>d cannot arise. It would imply that on B0 the values of

w(t), t~[1,d+l], could be chosen freely. Now for je[1,k] and

l~[1,k-j+l] there holds i~+l<i)+k-j+2<_i'j+l+k-j+l<...<i~+2<d+l , and hence there

is a w0E/~ 0 with wo(ij+l):=O, j~[1,k], l~[1,k-j+l], and wo(d+l)=wo(i~+2)= wo(i~_1+3 ) . . . . . Wo(i't+k+l):=l. From the structure of Hi(wo) it follows that

rank(Hi(wo))=c' and hence that gen. on 130 rank(Hi(w))=c'. This contradicts

that the last row in Ht'(w) is not gen. linearly independent from the

foregoing ones.

(iii-3) Next suppose that n'<_i~. Then consider (R'd,I",T") defined by

~n - n ~ . o , i - I r - F ~ . r i r R0:=s Ro, T":=T-n', and [ .=[0,T -1 ] \{h -n , z2 -n , . . . , zk -n ,d-n }. Note

that R~ satisfies (C1) and that Hl"(w ) satisfies (C2) on Br"(R~) , as Hz'(w)

satisfies this condition on B 0. As moreover d(Rg)+c(Rg)=

d-n'+(k+l)<_d-n'+c<_T-n'=T" it follows from steps (i) and (ii) that there is an

R" such that BT'(R'~)CBT"(R")EB~'(d-n',I" ) and hence Eo=I3T(RO)CI3T(a~'R')~

B (d , I ' ) c2 (d,I) , as c(a R")+d(a R')<_k+l+d<d+c<_T. Hence the desired results

follow.

(i~i-4) Finally suppose that i~<n'<_d. Let /~[2,k+1] be such that i~_l<n'<i[,

where i~+fi=d. Let H(w)~R (k÷l)x(k÷l) consist of the first k+l columns

of ttz'(w) and let H(w)~R (k-z÷~)×(k-:+:) be defined by

Chapter III 239

~(v):=

w(i~+l) ... w ( i i+k+l )

w(i~+lTl) ... w ( i i + l + k + l )

w(i~+l) ... w(i~+k+l)

w(d+ l ) ... w(d+k+l )

Note that the assumption that w~IV(d,I') is not gen. satisfied on B 0 implies

that det(H(w})-=0 on B 0. Moreover on B 0 there holds that, for given values of

w(t) for te[n'+l,T], the values of w(t) for te[1,n'] can be chosen

arbitrarily. Let Wl[n,+l,T]el3Ol[n,+l,T ] be arbitrary and fixed, choose w(t):=0

for te[1,n']\{i~_j+j;je[1,l-1]}~, and consider det(H(w)) as a polynomial in

the variables {w(i~_j+j); je[1,l-1]}. Note that indeed for je[1,/-1] i'z_i+y"

<i~_l+l<_n'. It is a simple matter of explicitly writing out H(w) to prove that 1-1 ., .

det(H(w}) contains the term //jffilW(h_j+3) with coefficient det(H(w)). As

det(H(w))--0 it follows that also det(H(w))=0 on B 0.

Now for w e b o define ~:=w I[i'+~,d+k+lleR ~, where T:=d+k-l-i~+2. Further let

~0 := /30 [ , = /30[ Ro:=s-n'Ro=~O+~lS+...+~~ sn-l+s ~ with ~0#0 [il+l,d+k+l] [n '+ l ,n '+~] ~ n - 1 and g:=n-n'. Let ~':=k-/+2 and ('fl,'f2,...,'~.~_l,d):=(O,i~+l-i],...,i~-i~,d-i~) and

let _~:=[0,T-1]\{~'l,...,~'~._l,d }. Define .4 and £ in terms of /~0 and @ analogous

to the definition of A and x in terms of R0 and w respectively as indicated in

the notation in the beginning of the proof.

Note that Lt(w)=nT(@ ) for /30, that hence det(H~.(@))-0 on B0, that ~'+d=T,

and that /~u satisfies condition (C1). As det(HT(@))----0 on Bo it follows that

there is a row in HT(5 ) which gen. is linearly dependent on the foregoing

ones. This shows the existence of a d.'e{~'l,...,~'~%,d } such that for gen. w~/~0

@clV(d',I)cR ~. Using arguments similar to those of steps (iii-1) and (i) it

follows that there exist {~'~,...,~'~, d.'}c{~'~,...,~'~._~,d} and 51eR, jel l ,k] ,

~ =BoCBT(R)~B~ such that ,a =~j=lC~js ~. Let m=s -~j=~c~js ~, t h e n ]flOl [n,+l,n,+T ] T

as c(fi))+d(R)<~'+d=T. From this and shift invariance of B 0 it follows that

~ 0 C H T ( $ n R). Let R:=s'tR, then I3ocI3T(R), as i;>_n', and [O,T-1]\I(R)c

{ ~'1+i~,..., ~'~._1+i~, d+i~}={i~,i~+~,...,i~,d). It was supposed that condition

(C2) is satisfied for c' and I ' of step (iii-1). Hence necessarily d'=d and

d(r)=d. Hence BoCBT(R)eB (d , I )cB (d,I), which proves the desired results.

240 Appendix

This concludes the proof of lemma 3-20-1 and lemma 3-20-2. •

To prove lemma 3-20-3 we make use of a result stated in lemma 3-20-5 which

also plays a role in the proof of proposition 3-25. Let 0 ~ i 1 < $ 2 < . . . < i c _ 1

<d~T-c and

~(w):=

w(i l+l ) w(il+2) ... w(i l+T-d )

w(i2+ 1 ) w(i2+ 2 ) ... w( i2+T-d )

W(ic_l+l) W(ic_1+2) ... w( ic_ l+T-d )

L e m m a 3-20-5 If rank (M(w))<c-2 everywhere on G0, then there exists

R~O, [l(R)<_ie_l~ I(R)~[O,T-1]\{i1,...,ic_l} , such that BOCBT(R)E~3 T.

P r o o f o f lemma 3 -20-5

There is at least one row of 3/ which is not gen. linearly independent from the

foregoing ones, say row k0. Exactly analogous to the proof of lemma 3-20-1 and

lemma 3-20-2 it follows that this row then always is linearly dependent on the

foregoing ones and that there exist a k such that on B o ~o(i%+t)= kO-I ~k=l akw(zk+t), tE[1,T-d]. By shift invariance of B0 this then also holds true

n ik 0 ~k0-1 i k * on [1,T-i%]. Define K:=a --:'k=l aka , then 13OCI3T(R)EE3T, while [l(R)=i%<ic_l

and I(R)~[O,T-1]\{il,...,ie_l}. •

P r o o f o f l e m m a 3 - 2 0 - 3

Let [~Jo, and define M as before, with {il,...,ic_l,do}:=[O,T-1]\I.

We state that gen. on /30 rank(M(w))=c-1. For suppose this is not true,

then det(MMV)¢0 is not generic, so det(MMT)=o on B 0 and rank(M(w))<c-2 on 130.

By lemma 3-20-5 this would imply that there exists R¢0 such that

BoCBT(R)~q*(d',I'), where d'<<_ic_l<d o and I'~[O,T-1]\{i,...,icq }. Hence 60c

BT(R)cIV(d',I') and d'<do, which contradicts the definition of d 0. 0 * d o c - 1 j i k

Now suppose that BT(Rj)~PTWnB (d0,I), j=l ,2 . Let Rj(a)=a +~k=laka , J i i a :=(al,...,ac_l) ' j=1,2. Using the notation of lemma 3-20-1, this means that

(al,1)lI1(w) =(a2,1)Hz(w)=O, so (al-32)31(w)=0. As gen. on B o rank(M(w))=c-1, we

C h a p t e r III 241

0 * get gen. on B0 al=a2; hence RI=R2, i.e., gen. on B o PrwnB (d0,I) contains at

most one model. From lemma 3-20-2 and the discussion following that lemma we 0 * 0 *

know that gen. on B o ~T(R(do,I))EPTWNB (d0,1). So gen. on B 0 PTW(3B (d0,I)

consists of a singleton, i.e., BT(R(do,I)). •

This concludes the proof of theorem 3-20. •

P r o o f o f c o r o l l a r y 3-21

From theorem 3-20 we immediately conclude that if B0eB r then gen. on B 0 0 ~oePT w. To prove the corollary, due to theorem 3-20 it suffices to show that

{13ocBeBr, C(B)=C(Bo) } ~ {B=Bo}. This easily follows from lemma II.3-34 and the

fact that for R~=0 with d(R)<T-1 there holds C(BT(R))=[l(R ). •

P r o o f o f t h e o r e m 3-22

(i) Obvious from the definition of po.

(ii) Obvious from theorem 3-20. • w . 0 * * 8 (iii) Obviously Bpoc~poClm((PT)C~T,. , so it suffices to show that IRTC~p~ ,. This

is immediate from corollary 3-21.

(iv) Consider p0 with the action of p0 restricted to models in the set B t.

For monotonicity and shift invariance, consider the inclusion conditions

on t which involve modelling w[[1,t_ q or w[[2,t], and w[[t,t]. Assume

E=ET(R)e~] r with c(r)+[l(R)<t-1.

For the monotonicity condition, observe that w[[1,t_q~Bt_l(R)eB~_l and

w[D,t]eBt(R)cB* t. From corollary 3-21 it follows that gen. in web

P°~(w [D,t_I])=Bt_I(R) and P°(w ] D,t])=Et(R), and the condition

Bt(R)[[x,t_x]cBt_l(R ) is trivial. Note that in fact corol lary 3-21 only gives 0

that for example P,_l(W[[1,t_l])=Et_l(R) gen. in w[[1,t_l]eBt_l(R). That this

also holds true gen. in web can be derived from the fact that d(R)_<t-1 which

implies tha t there is a linear bijection w[D,t_fl-->w from Bt_I(R ) to B.

~ ( R ) k For the shift invariance condition, consider two cases for R=.~k=o aka.

If a0~0 , then according to propositions II.3-30(ii) and II.3-33(ii) w[[2,t]e

B[[2,t]=B[[1,t_l], while if a0=0 , then w[[2,t]~B[[2,t]=l~t_l(o-lR). Generically 0 0 - 1

in web Pt_l(W][2,t])=Bt_l(R) in the first case, P,_l(w][2,tl)=13t_t(a R) in

242 Appendix

the second case. The shift invariance condition is trivally satisfied in both

cases.

Concerning linearity, let BieB~, i=1,2, then gen. in (wl,w2)et3axl3 z 0 0 P,wi=Bi, i=1,2, while due to theorem 3-20 gen. for 13ePt(Wl+W2) /~1+/~2c/~, which

0 proves linearity. Note that theorem 3-20 in fact gives Bl+132cBeP,w gen. in

weSx+S2, but one easily proves that this then also gen. holds true in

(wl,w2)~13,xB2 with w:=wl+w 2. Finally we give an example which shows that p0 is not monotone, not shift

invariant, and not linear.

Example. Let T=20, R:=al°+ag+2aS+aT+os+2aS+a4+a3+2a2+a+l, so c(R)+[l(R)= 21,

and consider B:=132o(R ). Further define Rl:=(a-1)R=a1'+a9-aS+a6-aS+a3-a2-1 with

c(R,)+~t(R1)=19, and R "-to" l~R--o"11+lal°+aag+lo'7+30"6+lo'4+30"3+lo- 1 with 2.-~ -~ / - ~ ~ ~ ~ ~ ~ ~ -~ 0 0 c(R2)+ct(R2)=20. Then gen. in w=/3 P19(wl[1,19])=P19(wli2,2o])=Blg(R1) while

0 P2oW={I32o(RI), ~2o(R2)}. So pO is not monotone as 132o(Rz)][L19]C.B19(R1), and

pO is not shift invariant as B20(R2)¢B20(O-.R1). pO also is not linear, as for 0 0 0 BI=B2:=B and for generic (wx,w2)el3xl3 there holds P2owa=P2ow2=P2o(Wx+W2)=

{B2o(RI),B2o(R2)}, while e.g. 132o(Rt)+B2o(R1)=B2o(R1)gt132o(R2) as would be

required for linearity. []

This concludes the proof of theorem 3-22. •

P r o o f o f p r o p o s i t i o n 3-25

For BoeB T we have to prove that gen. in w~B o the following holds:

{ReL(w):={R'#O; d(R')<T-I and W6~T(R')e~3~'}} ~ {~DC~T(R)}.

Notat ion. We will use some of the lemmas and the notation introduced in the

proof of theorem 3-20. Further we define Ko:={(d,I); BocW(d,I)) and

L(w;d,I):={ReL(w); I(R)~I,a(R)=d}. Again we can restrict attention to the case

d+(T-#(I))<T, i.e., d<_#(I). []

Remark. According to lemma 3-20-1 gen. on S 0 L(w;d,I)=o if (d,I)~Ii o. So gen.

on /~0 L(w)=[.J{L(w;d,I); (d,I)eKo}. []

Chapter III 243

The following lemma is crucial in the proof of proposition 3-25.

Lemma 3-25 Let (d,I)~K o be fixed. Then there exist n>_0 and R(J)~R[s],

je[0,n], such that

(i) d=[l(R(°))>~l(R(1))>...>~l(R(")),I(R(J))~I,je[O,n];

(ii) BoCBT(R(J))forallje[O,n];

(iii) gen. in wEB o L(w;d,I)=spano{R (i), j~[O,n]}:={R; 3oLj~R, j~[0 ,n] ,

c~0#0 , such that R=Z~=occjR(J)}.

We first give an interpretation of this lemma, then give the proof of

proposition 3-25 using lemma 3-25, and finally prove lemma 3-25.

I n t e r p r e t a t i o n . Lemma 3-25 has the following interpretation. Let (d,I)eKo,

then by definition of K 0 for every w~B o there is a remarkable law R w with

W~Br(Rw)cB*(d,I ). Now the lemma states that gen. in B 0 the class of

unfalsified remarkable laws in 8*(d,I) is independent of web o and that this

class is spanned by a finite number of "basic laws" R (j) which moreover are

true laws for g0. []

Proposition 3-25 can be proved by using lemma 3-25 in the following way.

Generically on B o L(w)=U{L(w;d,I); (d,I)~Ko} , and as K0 is a finite set it

suffices to prove that for (d,I)eK o gen. in web o {R~L(w;d,I)} ~. {BoCBT(R)}.

According to lemma 3-25(ii), (iii), ReL(w; d,I) gen. is of the form

R=Z~.=oc~jR (j), (x0~0 , with BOCBT(R(J)), je[O,n]. Using lemma 3-25(i) this

implies that for WeBo [R(i)(a)w](t)=O, te[1,T-a(R(J))]~[1,T-d]. This implies

that [R(a)w](t)=O, t~[1,T-d], and hence weBT(R ) as cl(R)=d for c¢0~0. So

BoCl3T(R), which proves proposition 3-25.

Now finally we prove lemma 3-25.

Remark. First note that if do:=min{d; there exists I such that (d,I)eKo} then

it follows from lemma 3-20-3 that for (do,Io)dio L(w;do,Io) gen. is a

singleton BT(R(do,Io)), and according to lcmma 3-20-2 Yoclg,l.(R(do,Io)), which

proves lemma 3-25 for d 0. However, in general for (d,I)~K o L(w; d,I) need not

gen. be a singleton. As an example, let T=5, Bo:=Bs(a-1), then (d,I):=

244 Appendix

(2,{3,4})eK 0 and for all WEB o L(w; d,I)~{Rc,,c¢eR} where Rc,:=(a-c~)(a-1)= a2-(~+l)o+~. []

P r o o f o f lemma 3-25

We give the proof by construction. First we define R (j) and then we show that

these have the desired properties.

Part (i) and (ii). Let

...<ic_l<d (c+d<_T), and define

w ( i l + l )

w( i 2+ 1) III(w):=

(d,I)EKo, [O,T-1]\I={il,i2,...,ic_l,d), 0<ii<i2<

w( i t +2 ) ... w( i l+T-d ) w(i2+2) ... w(ie+T-d)

w ( i c - t + l ) w ( i e - l + 2 ) ... w ( i c_ l+T-d ) w(d+l ) w(d+2 ) ... w(T)

and let Mk(W ) consist of the first k rows of tlI(w ).

As BocW(d,I), lemma 3-20-2 implies that there exists R (°) such that

13OCBT( R( °))cI3* ( d,I ). n ~ c - I i k d

Now note that for K=~k=laka +aca , ac¢:O , a:=(al,...,ac) , there holds

{RcL(w; d,1)}~{aHi(w)=O }. If gen. on 13 o rank(Mc_l(w))=c-1, then gen. a is

unique up to a constant factor and hence gen. L(w; d,I)={c~R(°); 0#c~R} and

lemma 3-25 is shown with n=0.

So suppose that not gen. rank(Mc_x(w ))=c-l; hence not gem

det(Mc_l(w)Mc_l(w)T)#o, so det(Mc_l(w)Mc_l(w)T)--O and rank(Mc_l(W))<_c-2 on

~0-

Lemma 3-20-5 implies that there exists R' with 13ocBT(R' ) and [l(R')<ic_l< d=d(R(°)), I(R')~Iw{d). Let R (1) be such a law for which d(R (1)) is maximal.

Let d(R(1))=idl and I i : = [ O , T - 1 ] \ { i l , . . . , id l ) .

Now either gen. on /30 rank(Mavl(w))=dl-1 , in which case gen.

L(w; ial,I1)={c~R(1); O#c~R) and we stop, or rank(Mdx_l(w))<_dl-2 on B0. In the

latter case, using lemma 3-20-5~ we find a law R (z) of maximal degree in the

class of laws with I(R")~I1U{ial } such that 13oC13T(R" ). So BoCBT(R(2)). Let

~l(R(2))=id2<idl_l<idl=Cl(R(1)) and I2:=[O~T-1]\{il~...,id2 }. Going on in this way we find a number n_<c-1 such that for j~[0,n] there

exists R (j) with 13oc13T(R(J)), I(R(J))~I, d(R(J))<cl(R(J-1)), while for ian:=

Chapter III 245

d(R(n)), In:--[O,T-1]\{il,...,ian}, gen. on 13o rank(Mdn-l(W))=dn-1, so gen. on

/30 L(w; idn,In)={ccR(n); O:pecE~}.

In this way we have defined n, R (j), j~[0,n] , with [l=~(R(°))>[l(R(1))>...> d(R(n)), I(R(J))~I, and /3oc/3r(R(J)). This proves (i) and (ii).

Part (iii). If R=Z~.=o~jR (j), c¢0¢0 , then cl(R)=d, I(R)3I, and /30C/3T(R)eET, as

c(R)+[I(R)=T-#(I(R))+[I(R)<_T, so ReL(w;d,I). We now have to show tha t gen. on B 0

if R~L(w; d,I) then there exist (~k , kc[0,n], with c~0¢0 , such tha t

R=~=o~jR (j). Without loss of general i ty we assume R and R (j) to be monic,

je [0 ,n] .

So let R~L(w;d,I) be given. If R=R (°) then we are done, else define

Rl:={R-R(°)}.flo , with flo such tha t R 1 is monic. We state tha t gen. on B 0 there

exists je[1,n] such tha t [l(Rx)=[l(R(J)). For suppose this does not hold true,

then there exists ikeI\{[l(R(J)), j e [0 ,n]} such tha t in Hz(w ) row k is not gen.

l inearly independent of the foregoing ones; hence by lemma 3-20-1 it is always

l inearly dependent on them and lemma 3 -20 -2 implies tha t there exists Rik , ~l(Rik)=ik, I(Rik)~[O,T-1]\il,...,ik) , with BoCBT(R i ). Now i~.<~l(R (n)) is

impossible by definit ion of n, so [l(R(n))<ik<[l(R(°~). Let j be such tha t

~l(R(J))<ik<~l(R(J-1))=:dj_a, then this contradic ts the const ruct ion of R (j) as

being of maximal degree in the class of laws g such tha t I([~)~[O,T-1]\{ix,i2, ...,idj_l_l} and ~0C/3T(R).

So indeed gen. on B 0 there exists je[1,n] , say Jl, such tha t d(R1)=

d(R(Jl)). I f RI=R (jr), then stop, else define R2:={R1-R(Jt)}.fll , where fll is

such tha t R 2 is monic. Going on in this way we gen. reduce R to laws of lower

degree in the set {~(R(J)), je[1,n]}. The process gen. will end either if we

find a ke[1,n] such tha t Rk=R (jk), or if we get R k with a(Rk)=d(R(~)). As

rank(Mdn_l(W))=dn-1 gen. on /30, also gen. Rk=R (~) in the la t ter case.

In this way we conclude tha t gen. on /30 if ReL(w; d,I) then R=R(°)+/3~IR~= R(°)+flolR(Jl)+fl~lflxlR z and going on in this way we find c~j, j c [0 ,n] , with

C~o=l , such tha t R=E~=occjR (1).

This concludes the p roo f o f lemma 3-25 and hence o f proposi t ion 3-25. •

2 4 6 A p p e n d i x

P r o o f o f c o r o l l a r y 3 -26

According to proposition 3-25 gen. in we/30 Goc13T(R(w)) , so gen. W~GT(R(w))

and hence gen. P~.W=GT(R(w)), R(w):=GCD{R; ReL(w)). So gen. on B 0 if ReL(w) then

13oc13T(R) and P~,zo=13T(R(w) )c13T(R ). • P r o o f o f c o r o l l a r y 3-27

Let R:=GCD{R#0; d,(R)_<T-1 and BoC13r(R,)e~) , R(w):=GCD{R; /~eL(w)} From

corol lary 3-26 gen. on G 0 if R.eL(w) then 13oC13T(R)¢BT~ hence by using

proposit ion II.3-36(ii) gen. on 13o 13T(R)CBT(R(w))=P*rw. On the other hand, if

R~0, d(R)_<T-1 and GoCBT(R)eB~, , then on B0 R~L(w), so BT(R(w))CBT(R), which

implies GT(R(w))C13T(R), so gen. on 130 P~WCBT(R ). •

P r o o f o f theore m 3-28

(i) Obvious from the definition of P*.

(ii) For B~IR t let R(13):=GCD{R#O; d(R)<t-1 and 13c13t(R)cBt}.

Monotonicity. Let 130e~lT, we130, P*t_l(wltLt_ll)=13t_l, Pt(wl[1,tl)=13t, then

we have to prove that gen. 13t[tl,t_llc13t_l. If /3t_l=R t-1 then this is trivial,

so assume 13t_l~R t ' l . Then from corollary 3-27 we can conclude that, gen. on

13o, 13t-l=13t-l(Rt-1) and Gt=Bt(Rt) , where Rt_l:=R(Gol[Lt_l] ) and Rt:= R(13o 103])" Now if R is such that B o [[1,t_llC13t_x(R)eBt_l, then

Bol[2,tlcB ol[Lt_llcBt_l(R ) and hence 1301tl,t]c13t(R)~Bt as c(R)+d(g)<_t-l<_t, hence for these R Gt(Rt)c13t(R), cf. proposition II.3-36(ii). This implies

Gt(Rt)cl3t(Rt_l) , hence also gen. on 130 Btl[1,t_1]=13t(Rt)l[1,t_11 c

13t( Rt-1 ) I [1,t-1]=13t-l( Rt-1)=Gt-1 • Shift invariance. Let 13oeBT, WeGo, Pt_l(w[t2,tl)=Gt_l(R), 13'=Pt(wltLtl),

then we have to prove that gen. B'cGt(a.R }. From corol lary 3-27 it follows

that gen. on 13 o R=R(13ol[2,t]) and gen. 13"=Bt(R' ) where R'=R(BoI[1,t]). Now if

is such that 1301 ~ * ~ * [2,tlc13t_l(R)eBt_l, then Bol as [1,t]cGt(aR)d3t c(crR )+t~(a]~)=

c(P,)+~(R)+l<_t-l+l=t. So gen. 13'=13t(R')c13t(P. ) where R:=GCD{a/~#0; d(/~)___t-1 and G N . o [ [2,t]c13t-l( R )~Bt-1} =erR.

Lineaxity. P*dcxw)=P**(w) for ~#0 follows from L(cxw)=L(w), ¢x#O. Now let

(131,132)~B~, Bi=13T(Ri) , /=1,2, ( t 0 1 , W 2 ) e B 1 X 1 3 2 , * ' * " PTWl=13T(R ), PTWz=13T(R ),

PT(wl+wz)=13T(R), then we have to prove that gen. 13T(R')+BT(R")C13T(R).

C h a p t e r I I I 247

According to proposition II.3-36(i) BI+B2=BT(K) with K:=LCM(R,,R2). According

to corollary 3-27 gen. in wl+w2, hence also gen. in (Wl,W2)eBl×B2, there holds

R=R(BT(K)). Moreover, gen. R'=R(Bx) and R"=R(B2). Because BicBT(K), i=1,2, if

BT(K)CBT(R. ) then also BiCBT(R), i=1,2, so BT(R')CBT(R), BT(R")CBT(R), and

hence also BT(R')+BT(R")CBT(R). (iii) This is immediate from corollary 3-27. Note that for /~:=~T p~w=~T gen. in

we~ T.

(iv) Let V:={BT(R); R=GCD{RA, AeA} for some {RA, AeA} with d(Rx)<T-1 and

BT(R~,)eB~}. For every procedure B~TCB~Tcim(PT) , so it suffices to show

that VcB~ and that im(PT)cV. If BeV, then corollary 3-27 implies that gen. on

B PTW=B, hence BeB~ . If Be im(P~), then either s=eTev or 3weR T such that

B=BT(R(w)) , R(w)=GCD{R; Ret(w)}. Because for ReL(w) d(R)<T-1 and BT(R)e8 ~ it follows that BeV. •

P r o o f o f p r o p o s i t i o n 3-32

Let BocBr. As L(w)cL(w) we conclude from corollary 3-26 that gen. on B o

{ReL(W)} =~ {BoCBT(R)}. So gen. on Bo BoCBT(R(w)) and hence gen. ['~W=BT(R{w)) where R(w):=GCD{R;ReL(w) }.

Now define R:=GCD{/~'~0; d(R)<T-1 and BOCBT(R)eB~}. For every R~0,

d(R)<T-1 with BoCBT(R)e~ ~ there holds that on B 0 ReL(w), hence BT(R(w ))cBT(R ) and gen. PTWCBT(R). On the other hand gen. on B 0 for ReL(w) ~ ~* BocBr( R)~ST, hence

gen. BT(R)CBT(~(W)) and gen. BT(R)G['~w. •

P r o o f o f t h e o r e m 3-33

It is trivial that P* is exact. That it is truthful immediately follows from

proposition 3-32. Strong prudence is shown by means of proposition 3-32 in a

way exactly analogous to the proof of theorem 3-28(iv), along with the

characterization of im(,hT)=Bp~ =8~..prS Linearity is shown by using proposition

3-32 and the arguments in the proof of theorem 3-28(ii).

Hence it remains to show that P* is bilaterally monotone. Monotonicity

follows as in the proof of theorem 3-28(ii). Finally let B0eBT, then we have

to prove that gen. in weB 0 there holds aBT_t+2CBT_t+l, where BT_t+i:= N$ PT_t+i(Wl[t_i+l,T])~ i=1,2. According to proposition 3-32 gen. on B 0 BT_t+i=

248 Appendix

Br_t+i(ei) , with Ri:=GCD{R#O; d(R)<T-t+i-1 and Bo [[t-i+l,T]C~T_t+i(R)eB;_t+i} ,

i=1,2. Now if R is such that [l(R)<T-t and BOI[t,TICBT_t+~(~)e~_t+X, then

translation invariance of B o implies (see proposition II.3-30(ii)) that also

~o[[t-I,T-1]=B0[[t,TICBT-t+I(R'), hence 1301[t_l,TlCBT_t+2(R)eg~._t+2. From this

it follows that BT_t+2(R2)CBT_t+2(R1)~ hence gen. on /~o l~T-t+2CI3T-t+2(Rl),

especially aBT_t+2caBr_t+2(Rt)=BT_t+x(R~)=Br_t+ 1 gen. on /~0. •

P r o o f o f p r o p o s i t i o n 3 -34

For B0=~ T the result in (i) follows from the truthfulness of PT and PT. So let T - 1 k B0C:R T. For O~R=~,k=oak a ~R[a] let l(R):=min{k;ak~O }. It easily follows that

{I3T(R)e~T} ~. {I(R)=0} for R~0, d(R)<T-1.

Let Bo=BT(Ro)eB T. From corollary 3-27 and proposition 3-32 it follows

that gen. on G 0 P~w=I3T(R) and PTW=BT(R), where R:=GCD{R'~e0; d(R')_<T-1 and

13ocl3T(R')~B~} and R:=GCD{R'¢0; d(R')<_T-1 and 13oct3r(R )eBT}.

(i) Let B0e~T, so l(Ro)=O, and let R'#0, d(R )<_T-l, such that ~oCBT(R )~T. Then according to lemma II.3-34 there exists F' such that R'=F'R o. If

l:=l(R')=l(F')~O then define F' by F':=a-IF ' and R':=~"R o. So /~ocG(R')eB~ as

/(R')=0 and c(R')+[I(R')=c(R')+[I(R')-I<T-I<_T. Now R'=o~R ' and it follows that

R=GCD{R'~O, [I(R')_T-1 and BoCI3T(R')e~;}=GCD{R'¢O; d(R')<T-1 and 60cBT(R')e

B~}=R and hence gen. PTw=Prw.

(ii) Let 13oc~3T, B0¢~7- , so l(R0)>l. If d(R')_<T-1 and /3ocBT(R )~3T then there

exists F' such that R'=F'R o and hence I(R')>I, so t3T(R')~B ~. This implies

that gen. on B o I,(w)=O, as according to corollary 3-26 gen. on B 0 ~ * ~ * T {R~L(w)cL(w)} ~ {~oCI3T(R)~ST}. So gen. on 130 R(w)=0 and PTW=~ . •

P r o o f o f lemma 3 - 3 5

First suppose t h a t gER T+I has a realization of dimension n, i.e., there exist

g~R z+ and (A,B,C)~Rn×'~xRn×IxR l×n such that gC[[o,rl=g and S(ge)=S(A,B,C,go).

Consider the input u defined by u(0):=l and u(t):=0 for t~e0. The corresponding

output y in S(g e) is given by y(t)=0 for t<0 and y(t)=g t for t>0. As

(u,y)eS(A,B,C,go) it is easily seen that gt=CAt-lB, te[1,T]. Define ReR[s] as

the characteristic polynomial of A. Then [l(R)=n, R(A)=O, and hence ~ICBT(R ).

Next suppose that ~BT(R ) with d(R)=n. Let R=k~=orkSk , then without loss

Chapter III 249

of generality we may assume that rn=l. Define (A,B,C)c•n*n×Rnxlx• l×n by

B:=(gl,...,On) T, C:=(1,O,...,O) and A:=[°zn-1], where OeR n-l, I,,_, is the

identity matrix in R (n-1)×(n-1) , and L - )p:=(_r0,_rl,...,_rn_l). A direct

calculation gives that gt=CAt-lB, te[1,T]. Define 9eeRZ+ by 9eo:=g o and

g~:=CAt-IB for teN. It is easily seen that S(ge)=S(A,B,C,9o). tlence 9 has a

realization of dimension n. •

CHAPTER IV

P r o o f o f p r o p o s i t i o n 3-2

In this proof and the next one we use the following result.

Lemma 3-2 If H=Bnl¢2, BeB, and B has a minimal realization Bs(A,B,C,D):=

m+n+q (v,x,w)~B,(A,B,C,O)nlz }.

P r o o f o f l emma 3 - 2

m + n + q If (v,x,w)eB~(A,B,C,D)AI 2 , then wcl3olq=H. On the othcr hand, if well and

~ i r n + n (v,x,w)cBs(A,B,C,D), then it suffices to show that ~v,x)etz . According to

proposition II.3-22 (A,B,C,D) is perfectly observable. From this and the

linearity and shift invariance of B it follows that there exists a linear map

L:(Rq)n->R n such that for (v,x,w)eB~ there holds x(t)=L(w][t,t+n_l]), t~Z.

Hence xel~ i f wel q. Moreover, Dv=w-Cx and D is injective, see proposition

II.3-22, hence vel 2. •

Next we prove the proposition. The result for Bs 2 is contained in the 2 lenm~a. To show that He~ 2 has a Bi/s/o realization, let H=Bnl~, where Be~3

has a minimal input/s tate/output realization Bq~/o , i.e., l lB={(u,y)e

(RmxRq-m)z; 3x~(Rn) z such that [~:1= I~ ~1 I:] }' cf. corollary II.3-23. Then

L'ir221slo:=Vilslont 2 r 2 .m+n+(q-m) ~'" an 12-input/state/output realization of H. Indeed, .m+n+(q-m) [:]

if (u,x,y)~13i/s/on t 2 , then // ~13nl~=tI, while for w~H there holds,

m+n+(q-m) B s and Bi/~/o follow~'J by (u,x,y)~13i/s/onl 2 . The results for R 2 R 2

considering RH, where R denotes the time reverse operator. •

Chapter IV 251

P r o o f o f p r opos i t i on 3-3

First we state two results which will be useful in the sequel. Then we use

these results to prove the proposition. Finally we prove the lemmas.

~'IXn Lemma 3 -3 -1 If A~R , then {x~l'~; ax=Ax)={O}.

Lemma 3 - 3 - 2 Let HeB2 and let B* be the closure of H in (Rq) z with

respect to the topology of pointwise convergence. If H has a realization

/3](A,B,C,D) with (A,B) controllable, then Bs(A,B,C,D) is a realization of

/3".

To prove the proposition we now first show that in a realization

B~(A,B,C,D) of HoB 2 for which n is minimal (for fixed m) the pair (A,B) is

controllable. Let R:=im[B AB ... An-IB]cR n and &=dim(R). As ARcR and im(B)cR it

follows that there is a choice of basis in R n such that in this basis

A: I A'A3] and B=rB1]. So in a corresponding partition [x:] of x in this basis L 0 A2J L 0 J ~

there holds for (v,x,w)~B s that ax2=A2x2. From lena 3-3-i we conclude x2=O,

hence 13~(A1,B1,C,D ) also is a realization of It. As n is minimal it follows

that dim(R)=d=n.

Hence for any m there exist controllable realizations of H with n

minimal, for given number m of driving variables. According to lemma 3-3-2

these induce realizations of B*. According to proposition II.3-22 B* has a

minimal realization, which induces a realization of H, see lenuna 3-2 in the

proof of proposition 3-2. So a minimal realization of H exists, and

B~(A,B,C,D) is a minimal realization of H if and only if Bs(A,B,C,D ) is a

minimal realization of B*.

Now if a realization /32(A,B,C,D) is minimal for H, then (A,B) is

controllable, and Bs(A,B,C,D ) is a minimal realization of /3", hence (A,B,C,D) is perfectly observable and D ~njective. On the other hand, if in a

realization B~(A,B,C,D) (A,B,C,D) is perfectly observable, D injective and

(A,B) controllable, then it easily follows that (A B) is surjective, hence

Bs(A,B,C,D ) is a minimal realization of /3*. This implies that /3](A,B,C,D) is a

minimal realization of H.

252 Appendix

Finally we prove lemma 3-3-1 and lemma 3-3-2.

P r o o f o f lemma 3 - 3 - 1

For A--0 the result is trivial. Hence assume A~0. Let p(s):=det(sI-A) be the

characteristic polynomial of A. As p(A)=O it follows that for solutions of

ax=Ax there holds p(a)xi=O , i=l,...,n. To prove the lemma it suffices to show

that with V:={weCZ; p(a)w=O) there holds {weV, t~y_lw(t)12< co}=~ {w=0}. M m i M

Let p(s)=s k. Hi=l(S-Ai) , IAtl>_...>_lA, l>0, ke:Y+, Ai~A j for i#j, Ei=xmi= n-k. Then dim(V)=n-k. Moreover one easily shows by induction on j that

the trajectories wij(t):=tJA~, j=O,...,mi-1 , /=I,...,M, give n-k independent

solutions in V. Hence weV if and only if there exist ~ijEC: j=l,...,mi, ! m i " t such that w(g)=~=ISj=lOQjtJ-1Ai . NOW consider such i=l,...,M, a with

t~zlw(t)12< ~ . If w#0 then let i+:=max{i; 3j such that c¢o-#0 } and i_:=min{i; 3j

such that cqj#0}. Taking t -> +co it can be shown that IAi_l<l and taking t -> -oo

it can be shown that [AI+I-~<I, hence I Ai-l<[Ai+l while i+>_i_. This

contradicts the ordering of the A's. Hence w=O. •

P r o o f o f lemma 3 - 3 - 2

If in Bs:=Bs(A,B,C,D ) the pair (A,B) is controllable, then for any to,tlJ 2 2

with t0<t I there holds Bs [[to,tl]=B2s[[to,tl], where Bs:=Bs(A,B,C,D ). Indeed, let (v,x,w)eBs and define (v',x',w') as follows. Let

(v',x',w')(t):=O for teT/\[to-n,tl+n], choose v' on [to-n,to-I ] such that

[B...An-IB] • col(v'(to-1),...,v'(to-n))=x(to) and on [tl+l,tl+n ] such

that [B...An-IB] • col(v'(tl+n),...v'(tl+l))=-AnX(tl+l), while v'l[to,tl]:=

v[[to,tl]. Compute (x',w')]to_n, tl+n ] according to = D "

follows that (v',x',w')eBs, and as it has compact support it is in B2s, while

(V ' ,X ' , 'W' ) l [ t O , t l ] = ( V , X , W ) [[ to , t l ].

Now let H have a realization B~ with (A,B) controllable, then we have to

show that B*=B':={w; 3(v,x) such that (v,x,w)eB~}. First let weB', (v,x,w)eBs. We conclude from the foregoing that for all

-co<t0Xtl<+cc so especially wl[to, tlleH][to, tllC ~*l[to, tl]. Completeness of B* implies that weB*, so B'cB*. That B*cB' is seen

as follows. According to theorem II.3-21 B'eB. Moreover, as B~cBs, it follows

Chapter IV 253

that ~3'oH. As /3" is the closure of H with respect to the topology of pointwise

convergence it follows from proposition II .3-3 that B'c/3 ' . •

This concludes the proof of proposition 3-3. •

P r o o f o f c o r o l l a r y 3 - 4

(i) Let Hc~32 have a minimal realization B2s(A,B,C,D), so according to

proposit ion 3-3 the pair (A,B) is controllable. Then Bs(A,B,C,D ) is a

minimal realization of B* as defined in lemma 3 -3 -2 in the proof of

proposit ion 3-3. That B* is controllable follows along the lines of the first

par t of the proof of lemma 3-3-2. Finally H=13*nlqz~ as for H=Bnlq2, B~B, there

holds HoB*c~3, so HcB*nlqgc/3nl~=H.

(ii) This follows along the lines of the first par t of the proof of lemma

3-3-2 , using the fact that for minimal realizations Bs(A,B,C,D) of BeB c

there holds that (A,B) is controllable, see Willems [74, section 4.8.2].

(iii) In the proof of proposition 3-3 we derived that B2s(A,B,C,D) is a minimal

realization of H if and only if Bs(A,B,C,D) is a minimal realization of

B* as defined in lemma 3-3-2. Hence the result follows from proposition

II.3-25. •

P r o o f o f p r o p o s i t i o n 3 - 5

Let ,.,ilslo:=ltu,x,y)e~2 ; y = 5 } be a minimal input /s ta te /

output realization of H~B2 such that //H={ ; 3x~l 2 such that (u,x,y)~Bi/s/o}.

It follows from proposition 3-3 that then (A,B) is controllable and that (A,C)

is observable, i.e., co1(C,~:,4,...,C.4 n-l) is injective. The last s tatement

follows from proposition 3-3 and the definition of perfect observabil i ty in

section II.3.3, which implies that {(u,Y) l[0,n_ll=0} ~ {x(0)=0}. 2

In Bi/s/o there exists a linear map L:u-~ x, as for u=O it follows from

lemma 3-3-1 in the proof of proposition 3-3 that x=O. As y=Cx+Du there exists

a linear map F:u--> y in H. So the remaining questions are in which case ud2

can be chosen arbi t rar i ly and in which cases F is causal or anticausal.

First we prove the implication ( ~ ) for (i), (ii) and (iii). Note that

2 5 4 A p p e n d i x

A+ o o ]

there is a choice of basis for ~n such that in this basis ~,=|0 A_ 0 |w i th L o o ~oJ

o-(A+)cC÷, a(A_)cC_, a(Ao)cCo, e .g , for .4 in real Jordan form. Let x= x_ , k ~ o J

B = B_ and C=(C+ C_ Co) be corresponding partitions. 0 If a(A)n%=O, then for uel~ let x÷,x_ be defined by x+(t):=

co k - I ¢o -1 1¢+1 t I:k= 1 A+ B+u(t-k) and x_(t):=-Sk=o(A_ ) B_u( +k), and let y:=C+x++C_x_+Du. It . ° [x÷l

e a s i l y follows that (x+,x_)el 2 and that : , _ / = / 0 A_ "- so "+]L,_j

13ilsl o. Hence u~l 2 is free. Moreover, if a(A)cC+ then the map L:u-> x

where x=x+ is causal, hence F:u-> y is causal. Similarly, if a(A)c£_ then

L:u --> x is anticausal, hence F:u -~ y also.

Next we prove (4) (~) . Suppose ,~eo{A) with IAl=l, say .Tlv=)~v where Ilvlt=l.

I t suffices to show that there is an uel'2 ~ such that there exists no xEl~ with

ax=~4x+Bu. Then there is no yel2 q-m with (u,y)dIH, as due to observabil i ty

{(u,y)e:~+(q-~)}~ {xe~. We then conclude that u is not free in l~ and hence

that there is no F:l~ --, l q-m with IIH=gr(F).

To construct u such that {x~l~; crx=.7tx+Bu}=O~ let al,a2e(Rm) n be such that

[An-lB...B]al=Re(v) and [A'~-IB...B]%=Im(v). This is possible as (A,B) is

controllable. Define ui~lr~ by uil[_n,_fl:=a i and ui(t):=O for tct[-n,-1], i=1,2.

Define Xi:={xe(Rn)z; ax=~4x+Bui} , i=1,2, and Xo:={x~(~'~)z; ax=Ax}. Then

Xi=xi+Xo, where x i is defined as that element xieX i for which xil(_oo,_nl=O,

i=1,2. It is easily seen that xl(t)=Re(Atv) and x2(t)=Im(Atv) for t>0. As

IAI=I it follows that at least one of the series YY I[xl(t)ll 2 and ~ [Ix2(t)l] 2 t > 0 t > 0

diverges, say t~ollXl(t)ll2=~. We now prove that then Xlnl'~=o. Let xeX1, i.e.,

x=Yc+x 1 with a~:=,4~. In an appropr ia te basis .7t=diag(A+o,A+l,A_,Ao) with

o(A+o)=(0}, a(A+l)cC+\(O}, ~r(A_)cC_, and a(A0)cC 0. Let ~=co1(~+o,$+1,~_,~0) be

a corresponding part i t ion of ~. I t easily follows that Yc+0=0. Further, in

order that E IIx(t)ll2<c¢ it is clearly necessary that ~:_=0, while E Ilx(t)ll2<~ t > 0 t_<O

implies that ~+a=0 and ~0=0 (note that x~(t)=0 for t<_-n), tlence if xJ~ , then

~=0. However, xx¢l'~, which proves the desired result

Finally we prove (ii) (~ ) and (iii) (~) . Suppose a(A}nC_#O, then we show

that there is (u ,y)e/ /H with ul(_~o,01 =0 but y](_=,o]#0. Note that

controllabil i ty implies that (A_,B_) is controllable and tha t perfect

observabil i ty implies that C_¢0. Let aeRa and actker(C_), and let b be such n - I n+ l m

that [A_ B .... B_]b=A_ a. Define uel 2 by ul[~,nl=b and u(t)=0 for t~[1,n].

Chapter IV" 255

-1 k+l Let y(t):=Du(t)-C_~=o(A_ ) B_u(t+k). Then one easily verifies that (u,y)~IIH and y(0)#0. This proves (ii) (~-). The proof of (iii) (=~) is completely

analogous. •

P r o o f o f p r o p o s i t i o n 4-1

We only prove (iii)+, as (iii)_ follows in an analogous way and (i)±, (ii)± are trivial. For (iii)+ we need to prove that @oH can be obtained as /z-limit

of t rajectories in H with left compact support, i.e., for any ~>0 there should

be a kc:~ and ~c(a*)kH+ with [[@_~[[2<~.

Let B~/8/o (A,B,C,D) be a minimal input /s ta te /output realization of H and

let :=//~, ('U,,X,y)~i/s/o. Define a linear map L:Rn* (R") n as follows. Let

{el,...,en} be a basis of N~ and let aie(R") n be such that [An-lB...B]ai=ei, i=l,...,n. Such a i exist as (A,B) is controllable. Define Lei:=ai and for xeR n,

X=~?=lXiei, let Lx:=~?=lXia i. Let 3I:=[]LT(I+FTF)LI[ 2 with FER n(q-m)xnm defined by

F : = n-2

CA B " C B

Let T~Z be such that tZ<T[[@(t)[[2<~e and [[Yc(T)I[2<e/(4M). Define (g,~,~)e 2

~ i / s / o b y (g,X,y)[(_m,T_n_l]:=0, g[[T_n,T_aI:=LYc(T), and U[[T,00):=~.[[T,00). Then

clearly ~ ( T ) = ~ ( T ) a n d hence (~,~,~)[[T,~)=(~,~,~)[[T,®). Define ~ : = H I ~ , then

We(a*)T-nn+ and liW--~ll2=t~<Tl[~(t)-~J(t)[12<_ 2(t~<TI]~z(t)l[2+t~<Tl[~)(t)[[2)<l~+ 2([[LYc(T)[12+[[FLYc(T)[[2)<_~+2M[[Yc(T)][2<e, as desired. •

P r o o f o f p r o p o s i t i o n 4 - 4

m q n I°:l lI:l 2 Consider H :={(v,w)el2xl2; 3xel 2 such that = }. Then B s is a minimal

input / s ta te /output realization of H'. If a(A)nCo=O then according to

proposition 3-5(i) there exists L:v-->w such that H'=gr(L), hence H=im(L), which proves (i). The results in (ii) follow from proposition 3-5(ii) and

(iii). Finally, (iii) follows from corol lary 3-4(iii) and the fact that (A,B) is controllable, see proposition 3-3. Indeed, it is a well-known result from

linear control theory that in this case det(sI-(A+BF)) may be any monic (real)

polynomial of degree n, by appropr ia te choice of F. See e.g. Kailath [33,

256 Appendix

section 7.1]. •

P r o o f o f p r o p o s i t i o n 4 - 6

In the proof of (ii) we use some results on the solutions of the discrete time

Lyapunov equation. For similar and more general results for the continuous

time case we refer to Glover [17, theorem 3.3], and Kailath [33, section

2.6.2]. We will first s ta te and prove a lemma which subsequently is used to

prove the proposition.

Lerama 4 - 6 Let (A,C)eRn×nxR q×n be observable, i.e., col(C,...,CA n-l) is

injective. Consider the discrete time Lyapunov equation ATKA+CTC=K. (i) I f there exists a nonsingular solution K, then a(A)nCo=O. (if) If a(A)cC+, then there exists a unique solution K and moreover K>0;

if a(A)cC_, then there exists a unique solution K and moreover K<0.

P r o o f o f l emma 4 - 6

(i) I f ATKA+CTC=K, then by repeatedly applying this identity we conclude that

K=(AT)nKAn+~-~(AT)t CrCA t, so K-(AT)nKAn>O as (A,C) is Observable. Now

suppose a(A)nC0%¢~, Ax=Ax with [A[=I and x=xl+ix2¢O , Xl,X2~R n. Then

x*(K-(AT)nKAn)x= (1-1A[2n)x*Kx=O, hence (K-(AT)nKAn)xi=O, i=1,2, which is in

contradict ion with K_(AT)nKAn>O. (fi) Let a(A)cC+ and suppose ATKiA+cTc=Ki, i=1,2. Then with Ko:=K1-K 2 it

follows that Ko=ATKoA=(AT)T~KoAn --> 0 for n -> oo, hence K0=0. Moreover, K:= Z~=o(AT)tcTcAt>O clearly is a solution. Analogously, if a(A)cC_ then K:=

-St=o[(A-1)T] t÷l cTc(A-1)t+I<O clearly is a solution, and it is the only one,

as for ATKiA+cTc=Ki, i=1,2, it follows that for Ko:=K1-K 2 Ko=(A-1)TIfoA-I=[(A-1)T]nKo(A-1)n-> 0 for n -> c¢, hence K0=0. •

(i) We now first prove proposition 4-6(i). As in the proof of proposition

II .3-22 we use for ze(Rd) z the notat ion z--:=zl(_~_l] , z+:=z][0,®). To

prove (~ ) , let be ( °~ m ' 2 0Iq

(v,x,w)~B2s there holds [laxl[K+l[w[[ =I[X[II~+I[v[[, hence IIw[12=l[vl[ 2. To prove (=~),

let Xo~R n and 13~(Xo):={(v,x,w)~B~; x(0)=x0}. Controllability implies that there

Chapter IV 257

is (v°,x°,w°)eB](xo) with (v°,x°,w°)l[n,~)=O. Analogous to part of the proof

of proposition 4-1 it is easily seen that we can take v°l[o,n_ll=LXo for a

linear map L:R n -~ (Rm) n, hence w°l[0,n_xl=Lx0 for a linear map L:R n -~ (Rq) n.

Define K:=Lr[,-LTL. Because Ilw°ll2=llv°lS 2 we conclude that ]](v°)--H2-H(w°)--I]2= n - 1 0 Et=o{llw (t)llu-IIv°(t)ll2}=xTKxo. As xocR n is arbitrary due to controllability,

if follows that K=K T is uniquely defined by By. Moreover, for any 2 (v,x,w)~B~(xo) there exists (v°,x°,w°)eB] with (v,x,w)--=(v°,x°,w°) -- and

(v°,x°,w °) I[n,00=O, due to controllability. Hence IIv--112-11w--ll2=xToKxo . 2 K is nonsingular, as for Kx0=O it follows that for (v,x,w)eBs(xo) with

v+=O there holds IIv--ll=Hw--II, hence w+=O and x0=O due to perfect

observability. We finally show K0 /,'0 ) Pontryagin

isometry. Let (a,b)~NnxN m and Ic ] : - - . D u e to controllability there 2 exists a (v,x,w)~B~ with (x(O),v(O),x(1),w(O))=(a,b,c,d). As (av,~rx,aw)eB~(c)

we conclude from the foregoing that Ilcll~=ll(~v)--112-11(a~ol--112=llv--112-11w--112+ IIbII2-11dlI2=llall~+llbll2-11dll ~, hence Ilcll~+lldll2=llall~+llbll 2 which is the desired

result.

(ii-1) Suppose (i) is satisfied, then K satisfies ATKA+CTC=K. Indeed, let x o 2 ~ x ( i ) ] = be arbitrary, then take (v,x,w)eBs(Xo) with v(0)=0, so (.w(o)j

[A B| [:0], and [A~] being an isometry implies IIAxoll~+llCxoll2=llXo]l~,., i.e.,

xTo(ATKA+cTc-K)xo=O. Now perfect observability of (A,B,C,D) implies

observability of (A,C), which is seen by taking (v,x,w)et3~ with

(v,w)l[0,n_l]=0. As K is nonsingular, it follows from lemma 4-6(i) that

a(A)nCo=¢) and from proposition 3-5(i) that L(A,B,C,D) exists. That it is an

isometry follows from (i). This proves (ii-1).

(ii-2) To prove (ii-2), note that in(K) does not depend upon a choice of A+ 0 coordinates in g~,. Choose these in such a way that A=r 1 with

r -, Lo a_J a(A+)cC+ and a(A_)cC_. Let C=(C+,C_) and K=| K+ K+_| be correspoding

1 T T K,-TJ T

partitions of C and K. As A KA+C C=K it follows that A+K+A++C+C+=K+ and

ATK_A_+CT_C_=K_. Moreover, (A+,C+) and (A_,C_) are both observable. From lemma

4-6(ii) we conclude that K+>0 and K_<0. From this we easily get (ii-2).

(ii-3) This result is an immediate consequence of (ii-2) and proposition

3-5(ii) and (iii).•

25S Append ix

P r o o f of p ropos i t ion 4 - 7

M'= [A+BF Be] ~R(,+q)×(,+m) SeR,×, invertible and h:=hjTeR "×" invertible. It Let " I C+DF DRJ . s o s-~n . rio Ro

is easily shown that I IM I l ,s a ( I l ' { l ) Pontryagin isometry k 0 IqA k 0 Ira) t. 0 Imj t. 0 Iq9 f __ "~ f L _ %

• . . . if i and only I f M is a ( L J U q J ° I m ' o ) Pontryagm lsometry, where K : = S r k S . IIence

it suffices to prove the proposition for the case S=I. f %

Now hi is a ( ~ ,,,~oz ' ~ ~°zq

(a,b)eNnx, m and / : / :=MIb/ there holds ~ J ~ J

0 = . ( c l T ~ K o ~alTCXIIXI2 l kb3 kx2, x22J ' LbJ ko

where a direct calculation shows that Xn=(A+BF)TK(A+BF)+(C+DF)T(C+DF)~ X22=RT(BTKB+DTD)R, and Xel=xT2=RT[BTK(A+BF)+ DT(C+DF)]. Hence M is such an

isometry if and only if Xll---K, X22=Im and X12=0. Now X22=I,n is equivalent with

(R) as R is invertible, and then X12=0 is equivalent with (F) as R is

invertible and BTKB+DTD also, according to (R). Given (F) and (R), X11=K is

equivalent with (ARE) by using the expression (F) for F. •

P r o o f o f lemma 4-8

According to proposition 3-3 (A,B,C,D) is perfectly observable and (A,B)

controllable, hence stabilizable, see Kailath [33, section 7.1]. The result

follows immediatcly from Payne and Silverman [57, theorem 2.1 and lemma

, t .5} . •

P r o o f o f theorem 4-9

According to propositions 4-6 and 4-7 L:=L(A+,B+,C+,D+) is a causal, time

invariant isometry with im(L)=H. According to corollary 4-2 it suffices to

show that H+=LI2(Z+,~m), as L+ is unique (up to isomorphisms of ~"). As L is m 2 causal LI2(~_+,R )oH+. It remains to show that for (v,x,w)eB~(A+,B+,C+,D+) with

w--=O there holds v--=O. From corollary 3-4(iii) and proposition 3-3 it

follows that (A+,B+,C+,D+) is perfectly observable and that D+ is injective.

[cA+B+]I; ] it follws by induction that (v,x)(t)=O for t=-n+l , . . . , -1 , hence + D+J

Chapter IV 259

v = 0 . •

P r o o f o f p ropos i t i on 4-11

Let [.+ be the forward scattering representation of H and let L:=I~L+~. Then L

clearly is isometric and time invariant and im(L)=~im(L+)=~t=H. According to

corollary 4-2 it now suffices to show that LI2(7/_,Rm)=H_ in order to conclude

that L=L_. Define 12(7]++,Rm):={ve12(71,Rm); v(t)=O for t<O}=a*t2(7/ +,R"), then L12(7/_,Rm)=RL+I2(7]++, R m) = RL+a*I2(7/+, R m) = Rcr*L+I2(Z+, R m) = ~a*H+

=aR~I+= a{weH; w(t)=O for t>O)={weH; w(t)=O for t>O}=II_. •

P r o o f o f c o r o l l a r y 5 - 3

It suffices to prove that P=L+L+=L_L*. We will prove P=L+L+, as the other

result follows analogously. P:/q-> I q is uniquely determined by the conditions

(i) P*=P; (ii) p2=p; (iii) im(P)=H, see e.g. Akhiezer and Glazman [2, section

31]. Now L+L: clearly satisfies (i), while (ii) follows from the fact that L+

is isometric, so L~L+=I. As im(L+)=H it only remains to show that Hcim(L+L*+). Let w~H, then w=L+L+w which is seen as follows. Let v be such that L+v=w, then

for any v' <w,L+v'>=<L+v,L+v'>=<v,v'> as L+ is isometric, so by definition

L+w=v and L+L+w=L+v=w. Hence Hclm(L+L+). •

P r o o f o f t heo rem 5-4

In this proof we use some standard results on the discrete Fourier transform

and its inverse, the Z-transform. We will state these results in a lemma, give

an outline of the proof of the lemma and finally prove the theorem by means of

this lemma.

Notat ion. Let C0:={zeC; ]z l=l } and for f:Co --> C d let llf]12:=

(27r)-lf[f(eie)]*f(eie)dO, where • denotes complex conjugate transpose. We - T r

define Ilfll%0} and on

With this inner product L d is a Hilbert space. Let

ld(c):={xe(Cd)Y-;t~z[x(t)]*x(t)<c¢ }. We define the discrete Fouriertransform

260 Appendix

F:L~ -->/~(C) by (Ff)(k):=f(k):= (2~r) -I j'~ e~kef(e~e)dO and the Z - t r a n s f o r m iO oo - i k O - Tr

Z:ld(C)->L~ by (Zx)(e):=~k=-co e x(k) These t ransformat ions are

wel l -def ined in a l imi t - in -mean sense. []

Lemma 5 - 4 (i) F is un i ta ry f rom L d to I~(C), i.e., it is isometric and

surjective; Z is the inverse of F.

(ii) Let Hal3 2 have minimal real izat ion B~(A,B,C,D) with a(A)nCo=O and

let L:=L(A,B,C,D) be the corresponding driving opera tor , then Z(Lv)= G.Z(v) with G(z):=D+C(zI-A)-IB, z~C0; if L is isometric then Z(v)=

G*" Z(Lv) where G*(z):=[G(z)]*.

(iii) For L as in (ii) IILIh:=sup{llLvll; Ilvll--1}--llGllco.

P r o o f o f l emma 5 - 4 (outline)

The result in (i) is the well-known Fourier-Plancherel theorem, cf. Kato [40,

section V.2.2].

To prove (ii), let a basis in R n be chosen such t ha t A= IA+ °I with L° J A~

< , a(A+)cC+, a(A_)cC_, and let B= B_ Then for vel'~ LV=LlV+L2v+Dv whereV 2 (Llv)(t):=C+S~=lAk+_lB+v(t_k) and (L2v)(t):=

m -1 k + l -C_Sk=o(A_ ) B v(t+k), cf. the p roof o f proposi t ion 3-5. By the definition , co - k k - 1 co k - 1 k + l

of Z - t r a n s f o r m we obta in Z(Lv)=[C+(~k=xz A+ )B+-C_(Ek=oZ (A_ ) )B_+D]Z(v) and as a(A+) and a(A -1) are contained in C+ there holds for zeC0 that S c o - k , k - 1 ~ c o k . _ - l , k + l - - - 1 , T - - - 1 , - I k=xz .% = z-I(I-z-IA+)-I=(zI-A+) -1 and Z~k=oZ (A_ ) =a_ (l--Z:a_ ) =

-(zI-A_) -1. From this we obta in G. If L is in addit ion isometric, then for

vel 2 with Lv=w there holds for a rb i t r a ry v' with Lv'=:w' tha t <G Zw-Zv,Zv >=

<Zw,GZv'>-<Zv,Zv'>=<Zw,Zw'>-<Zv,Zv'>=<w,w'>-<v,v'>=<v,L*Lv'>-<v,v'>=O as

L*L=I. tlence G*Zw= Zv as desired.

To prove (iii) note tha t for [Iv[I=1 ]lLvll--llGZ(v)ll<_llGl[~, hence it suffices

to p rove tha t fo r every e>0 there is a vcl~ with Ilvil=l and [ILvH>llGll~-e. As IIG(z)ll is continuous on C O which is compact it follows that there is a

80c[-rc,rr ] such that [[G(ei°°)]l=sup{][G(z)[[; zeCo}=:lIG[]oo. Let aeC m with IIall=l be

such tha t ]]G(eW°)all=llGI]co. Continuity of G(z) implies tha t there exists a 6>0

such that fo r I8-001<6 IIG(ei°)alI>lIG}lco-e. Then for 18+0a1<6 there holds

I[G(eie)all=llG(e-ie)all>llG[Ico-e. Now define uel'~ as follows. If 80=0 , then take

aeR m (this is possible), and let u(eiO):=(rr/a)l/2"a'I(_a,a)(O), where Iv(O)=I

Chapter IV 261

for OeV and 0 for O¢V. If 00~0, then take 6<1001 and let u(ci°):=(lr/26) v2.

{a.I(oo_8,Oo+a)(O)+~.I(_oo_6,_eo+~)(O)}. Let v:=Fu. Then []vl[=JluIl=l and vcl~,

i.e., v is real, as -~)=(2rc)-lf're-ikOu-((eiO)do=(2rc)-lf're-ikOu(e-iO)dO=v(k), - ~ - f t

keZ. Now IlLvlh=llaull which is easily seen to be larger than Ilall~-s. •

Theorem 5-4 is easily proved by means of this lemma. Let L:=L+L*+-L+L~* and

G:=G+G+-G+G+ , then for w~l q there holds Z(Lw)=Z(L+L~w)-Z(L~L~*w)= G+Z(L+w)-G+Z(L+ w)=G+G+Zw-G+G+ Zw=GZw, and hence ]]Lil2=sup{i]GZwil; ]]w]i=l}=liGi] ~ which follows from the proof of part (iii) of the lemma. The

result for G_ follows in an analogous way. •

P r o o f o f p ropos i t i on 5 - 5

We first state a preliminary result. Let L1,L2cR n be linear subspaces with

dim(L1)>dim(L2) , then LlnL~2~e{0}, which easily follows from dim(Ll)+dim(L~)=

dim( L1)+n-dim( L2 )>n. Now let (ra,n,m',n'):=(ra(H),n(H),m(H'),n(H')) and suppose m>ra'. It suffices

to prove that Hn(H')-L~{0}, as from O~weHn(H') ± we conclude g(H,H')=

HP-P']]>_ j~wlj]lPw-P'wl]=l, while g(H,H')<I always holds true.

According to corollary 3-4 and proposition 2-2 there exist controllable

systems B,B°e~ such that H=Bnl~ and H'=B'nl q, where (m(B),n(B),m(B'),n(B'))= (m,n,m',n'). For TeT]+ let B(T):={w~B; w(t)=O for t<0 and for t>_T}cH. We will

show that there is a T such that there exists O~wEB(T) with

W[[o,T_ll_I_I3'I[o,T_I]. As B'[[o,T_I]=II'I[o,T_I] this implies that O¢wcHn(tt') "l-, as desired.

Now take T such that (T-2n)m>n'+Tm',i.e., T>(n'+2nm)/(m-m'). As 6 is

controllable we conclude that in B(T) the m inputs can be chosen freely on

[n,T-n-1], hence dim(B(T)][o,r_q)>(T-2n)m. From a minimal input/state/output

realization of B' it is evident that dim(B'l[O,T_l])<_n'+Tm'. From the result

stated in the first lines of this proof it follows that there exists

O~:X~[B(T) I[o,T_q] n [13'I[o,T_l]] ±, Define w~13(r) by Wl[o,T_z]:=x and w(t):=O for t~[0,T-1]. Then O~weHn(H') ±. •

2 1 3 2 A p p e n d i x

P r o o f o f p r o p o s i t i o n 5 - 7

From proposi t ions 4 - 6 and 4 -7 charac te r iz ing isometries we conclude tha t 2 2 2 2 in IIx+(t+l)llK++llw(t)ll =I}x+(t)IIK++IIv+(t)I I . As x+e z this implies Ilx+(O)ll~+=

iiv~-112_llw--112= + 2 + 2 Ilw II-IIv+ll • From these proposi t ions and the fac t tha t /~B;= 2 B~(A_,B_,C_,D_) we conclude tha t IIx_(t-1)[l~++llw(t)ll2=llx_(t)ll~.++l[v_(~)ll =, so

IIx_(O)ll~+=llv+_+ ll2-1lw++ ll2. Next we p rove the express ions fo r Ilxoll++ and Ilxoll--- We use the facts

co k - 1 0 co k - I t ha t in B+s x+(O)=Sk=lA + B+v+(-k) and in B~ x_( )=Sk=IA_ B_v_(-k), t ha t L+

and L are isometries, t ha t ax+=x_ and all=H, to ob ta in that

Ilxoll~+ = inf{ Itv;-ll2; (v+,x+,~o)~B~ +, x+(0)=Xo} = inf{ IIv+ll2; (v+,x+,zo)et3+~,

x+(0)=xo} = inf{ Ilwll2; (v+,x+,zo)ez~ +, x+(O)=xo} = inf{ II~oll~; (v_,x_,v))~t~;, x_(-1)=xo} = inf{ 11~112; (v_,x_,w)et3;, x_(0)=xo} = inf{ IIv_l12; (v_,x_,w)ct~, x_(0)=xo} = inf{ IIv+_+lle; (v_,x_,~o)Et3 L x_(O)=xo}=llxoll2__ = inf{ Ilxoll_2+ll~o++llZ; ( v _ , x _ , ~ o ) ~ , x _ ( O ) = x o } = Ilxoll_ 2 + i n f { IIw÷l12; (v_,x_,w)~t~-~, x _ ( - 1 ) = X o } =

IIx0112_+inf{llvJ+ll2; (v+,x+,w)e/3+~, x÷(0)=xo} 2 2 . + 2 = Ilxoll-+llxoll++mf{ IIv+ll ; (v+,x+,~o)elS~, 2 2 2 r -1 x+(O)=xo}=l[Xol[++HXo[l_. I t remains to p rove tha t [IXoll++=xoQ+ x o.

7n This is a wel l -known result f rom l inear quadra t ic control . For v¢I 2 define

- - ~ k - 1 - - Tv :=~=~A+ B+v(-k), then for (v+,x+,w)~B+~ there holds Tv+ =x+(O). Let T * R n 2 . - - 2 : --> 12(N,R m) denote the adjoint o p e r a t o r of T, then IIx011++--lnf{llv + II;

Tv-+-=xo}=llT.(TT.)-lxoil2= T • -~ xo(TT ) xu, and a direct ca lcula t ion shows that

TT*=Q+. •

P r o o f o f l e m m a 5-8

Let B2s(A,B,C,D ) and RBs2(,4,B,C,D ) be minimal rea l iza t ions of HaB 2. A change of

cor rd ina tes x-~ x':=Sx leads to a t r ans fo rma t ion of p a r a m e t e r s (A,B,C,D)-~

(SAS-I,sB,CS-1,D) and (A,B,C,D)--> (S~tS-1,SB,CS-1,D). From (ARE) it follows ~ 1 / 2 ~ - 1 , , 1 / 2 r A2 , T

t ha t (K+,K+) -> ((S-I)TIi+S-I,(S-1)Tfi+S-1). Now let n + n + l~+=un ~ with U

o r thogona l and A=diag(A1,...,A,) , AI>_...>_An>0. A direct calcula t ion shows

t h a t S:=A-I/~uTK1/+ 2 gives the desired result . •

P r o o f o f p r o p o s i t i o n 5-9

We will show tha t a(,4)cC+. According to p ropos t ion 3-5( i ) there then exists a

C h a p t e r I V 2 6 3

2 ^ ^ ^ ^ ^ map v -~ ~ with domain l~ for Bs(A,B,C,D). As D=D+ is injective we can conclude

that m(~I)=coo(ft)=m. As B](A,B,C,D) is a realization of H we conclude that

To show that a{A)cC+ we use the fact that with Db:=D + (Ab,Bb,Cb, Db) are

the parameters of the forward scattering representation of H with the

corresponding solution Ii b of (APE) given by A=diag(A1,...,An). Define

A:=diag(A1,...,A~) and A2:=diag(A~+1,...,An). From propositions 4-6 and 4-7 we

conclude that I=BTABb+DTbDb from (Rb) , that o=BTAAb+D~Cb from (Fb) , and hence

from (AREb) that A=A~AAb+CTcb.

Take partitions as in step 3 of the algorithm for balancing. Let c~a(A)

atld Ax=c~x for some 0¢:x~C . Let x:= I q ~C • A direct computation of x*(AREb)X • T ^ T ^ L ° J 2 • ^

shows that then ~ [A2~A2A2~+C C]~=(1-1al )~ A~. As A>0 it follows that Ic<l_<l.

It suffices to show that Ic~l~l. As A2>0, Ic~l=l would imply that A21~=0 and

hence that AbX=~X , which contradicts the fact that a(Ab)CC+, cf. proposition

4-6. •

CHAPTER V

P r o o f o f p r o p o s i t i o n 2 - 8

The minimax p rope r ty is wel l -known and shown e.g. in Stewart [67, theorem 6.5

o f sect ion 6.6]. As the p rope r ty is o f crucial importance in the sequel we

give its p r o o f explicitly.

Let A have (SVD) A=UL-'V T. For xeR n2 and LcR n2 let y:=VTx and L':=VTL. Then

min{ max IIAxU. • . n2 2 2 vz n 2 2 z ~ i = l y i = l ~ dlm(L)>n2-k}=mm { max {Si_ 1 aiyi} ; dim(L')_>n z-

O~xeL IlXll ~ -- O;ayeL' - k}=:c¢ k. Taking L':=span{ek+x,...,en2 } shows tha t (xk<ak+ 1. On the o ther hand, if

dim(L')>_n2-k , then V:=L'nspan{e~,...ek+1}¢:{O}, hence V contains a y ' of norm 1.

This implies tha t (xk>ak+ 1.

Finally no te tha t c~ k is achieved for L':=span{%+l,..,en2}~ i.e., for

L =VL =span{vk+l,...,Vn2}=Ln2_ k. •

P r o o f o f l emma 2 - 9

If nl=n2, A=diag(dl,...,dn2 ) with dl>...>dn2>O a n d L=• n2, then it easily

follows tha t eA(R n2) is wel l -def ined and equal to (dl,...,d,~2). To prove the

general case, let LcR n2 with dim (L)=d and let the columns of BcR '~2×a form an

o r thonormal basis of L. Further let BTArAB have (SVD) UZU T with Z=

diag(al,...,ad) , al>...>_aa>O. Then it follows tha t fo r O~xeL, say x=BUy, there

holds uAxll_ IIZV2YlI--. From this and the fac t tha t uTBTBU=Id it is easily seen II x I] II y II d

tha t eA(L)][1,d]=el/2(R ) which was a l ready seen to be well-defined• Hence v2 2/2 0 n~ ¢A(L) is also wel l -def ined and is equal to (al~...,~d~ , .... ,. •

P r o o f o f p r o p o s i t i o n 2 - 1 0

If k<nz-r then eA(Lk)=0 and the results easily follow. We hence will assume

tha t k>n2-r. (i) As dHn(Ln(span{xl,...,Xj_l} ) )>_k-3+1 it follows f rom proposi t ion 2-8 that

fo r j=l , . . . ,d im(L) ei(L)>_an2_k+ j. For L~ it follows by induction tha t for

C h a p t e r V ' ) 6 5

j= l , . . . ,k ej(L~.)=a,2_k+ 2 (ii) First suppose that nl=n 2 and that A=diag(dl,...,dn2 ) with dl>_...>dn2_k>

dn2_k+l>_...>_dr>dr+ 1 . . . . . dn2=O. We then have to prove that eA(L)=(dn2_k+l, ...,dna,0,...,0 ) implies that L=L~, where L~=span{en2_k+l,...,e%}.

Let 0<61<...<6 ~ denote the distinct values in {dn2_k+l,...,dr} and let 6j have multiplicity mj, E~=lmj=max{O,r-n2+k ). Define Mj:=n2-r+ZJ=lmi, j e l l , s ] ,

and Mo:=na-r. We will show by induction that for j=O,...,s span{e~2_Mj+l,...,en2 } c L. For j=s this gives the desired result, as

n2-Ms+l=n2-k+l. First consider j=O. Proposition 2-8 implies that for

dim(L')>k+l el(L')>d . -k , hence dim(L)=k. As ei(L)=O for ie[r-n2+k+l,k ] there - - 2

is a subspace VocL with dim(V0) _> n2-r and Ax=O for xeV 0. lIence

Vo=span{er+l,...,en2 } which proves the result for j=O. Next suppose that

span{en2_Mj_l +i,...,cn 2}CL for some je[1,s]. As

eA(L ) s =(5:, ..,Sj, Sj_l,...,~j_l,...,~l,. ,~1,0, ,0), wimre ] [ ( E i = j + l m i ) + l , k ] -/ . . . . . .

0 appears nz-r and 8i appears m i times, i=l,. . . , j , it easily follows that

there is a subspace gjcL with dim(Vj)=mj, Vjxspan{e%_Mj_l+l,...,en2 } and I IAz l l .

0,~v~max ~llXll =o,.j Hence Vj=span{ena_Mj+l,...,en2_Mj_l } which concludes the

inductive proof.

We finally prove the result for general A. Let AeR "1×"2 have (SVI)) A=UEV T

and define D:=diag(cq,...,an2 ) where al>_...>_an2_k>an2_k+l>_...>_Crr>ar+ 1 . . . . . an2= 0. Suppose ¢A(L)=(crn2_k+l,...,a,,2,0,...,O), and let L':=vTL. With y:=VTx there holds IIAx II IIDyll IlXll -- I - ~ "° Using the or thogonal i ty of V we conclude that eA(L)=SD(L'). We have shown that hence L'=span{en2_k+l,...,e,=}, so L=VL'=L~. (iii) The proof of (~ ) is direct. To prove (=,), let ¢A(L)=(an2_k+l,...,

a t ,0 , . . . , 0 ) . Then dim(L)=k. Let the singular values of A satisfy

6rl>_...>_6rCl>O'Cl+l=...=tTn2~k~-~n2_k+l=...=~c2>Oc2+l~...>Un2~O. From the proof of

(ii) it easily follows that L':=span{v%+l,...,vn2}cL. Further there is a

subspace L"cL with L"±L', dim(L")=k-dim(L'), such that max ,,IIAzll O~ x e L IIXU ~" O'n2-k"

Hence L"c span{vq+l,...,v%}. •

P r o o f o f p r o p o s i t i o n 2-11

It follows from definition 2-2 that eD(d,a) = ItSa[I and from definition 2-4

that tD(d,M)=es(M±), which is well-defined according to lemma 2-9. •

2 6 6 A p p e n d i x

P r o o f o f p r o p o s i t i o n 2-12

The ordering of the misfits is lexicographical, cf. definition 2-5, and

according to proposition 2-11 eD(d,M)=es(M ±) with S the empirical covariance

matr ix corresponding to d. This enables us to use proposition 2-10.

(i) This is trivial from the definition of P~cto ~. (ii) Let L :=span{xl,...,xw}. As rank(S)=r dlm(L )=r<_cto b and eD(d,L*)=

es((L*)Z)=0, hence minimal. To show optimality of L* it remains to prove

that {dim(M)<r, eD(d,M)=O) ~ {M=L*}. If ¢D(d,M)=es(M-t')=O then ]]Sal]=O for all

aeM ±, so M&cker(S) and hence M~im(S)=L*. If in addition dim(M)<r=dim(L*) then

M=L*. * ± g~ , . •

(iii) First note that (MCtot) =Ln_ctot as defined m section 2.1.2. Let /lle~

with cD(M)<Ctol, ± " " " _ then dim(M )>n-cto I. According to proposition 2-10(i) D 2_ * D * • •

then e (d,31)=es(31)>es(Ln_ctot)=e (d,Mctot), while according to proposition

2-10(ii) aCtol>aetol+l lmphes that es(M )=¢s(Ln_ctol ) if and only if

1V±I =Z*n_Cto l ~ i.e., M=M:tol. (iv) As for cD(M)<_cta there holds dim(M±)>n-ctoz and ~rctol>__~rr>O we conclude

from proposition 2-10(i) and (iii) that M has minimal misfit if and only

if MX=L'+L '', where L':=span{ufi aj<aCtol } with dim(L')=n-Cl-dim(M(Crctol)) and

L"cM ( ff etot ) with dim( L")=n-Ctol-dim( L')=crCtol-dim( M ( actol) ). Then M=

(L')±n(L")±=M*~I+L, where L:=M(acto~)n(L")± has dimension dim(M(aCtot))- dim( L")=Ctol-C 1. []

P r o o f o f p r o p o s i t i o n 2 - 1 3

D to l (i) This is evident as el(d,{O})=crx< Q .

(ii) Let L :=span{xl,...,XN} , then el(a,L )=0<e 1 . If cD(M)<r then dim(M±)_>

cl(d,bl)>_ar>e 1 . So the n-r+l and according to proposition 2-10(i) D tot

minimal achievable complexity is r. To show optimality of L* it remains to

prove that {c°(Pl)=r, eD(d,M)=O} =~ {M=L*}, for which we refer to the last part

of the proof of proposition 2-12(ii). "M *'-1- L* D • (iii) First note that ( k ) = n-k as defined in section 2.1.2, hence e (d,Mk)=

• D * to l ea(Ln_k)=(O'k+l,..,O'r,O,...,O), especially ei(d,31k)<<_ak+l<el for all

i~[1,n]• If cD(M)<k then dim(Ml)>n-k+l and according to proposition 2-10(i) D to l el(d,M)>ak> Q . Hence the minimal achievable complexity is k. As ak>ak+l we

C h a p t e r V 267

conclude from proposit ion 2-10(i) and (ii) that among models with cD(31)=k eD(d,M) is uniquely minimized by taking N:=31~. •

P r o o f o f p r o p o s i t i o n 2 - 1 8

Let the data be generic, so Sxx a n d Syy are invertible. Let MeN and N~:=

{a2eRn2; 3aleR nl such that (ax,a2)eM-L}. Suppose that (al,az)~M 1- and

(a{,a2)~M3- , then (ax-a[,O)eM 1- and as the projection in M on the first n 1

coordinates is surjective it follows that al=a ~. Hence there exists a linear

map A: Rn2-> R 'q such that ML={(at,a2); a2eMl2, al=Aa2}. For a=(al,a2)eM3- let

V2 . P T Sxx Sxy IIQV2~II cc=S. .a2 , then it follows that e (d,a)={a [ ] .1/2.¢ T.~ 1/:~ at /ta2~yya2.~ = where _ . . . . r . . . . . . , / , . L S y x %) I, {2: = y y ( A b'xxA+ b'yy--}-A ,.~'xy+b'yxA)byy. A S Ot /O¢ l I a n d only i f a2_l_(2)a ~ i t f o l l o w s

p 1/2 £ from definition 2-17 that for generic data e (d,3I)=eO1/2(Syy312) which is

well-defined according to lemma 2-9. It

P r o o f o f l emma 2 - 1 9

Let L2cR n2 be given and MeN(L2). In the proof of proposit ion 2-18 it was shown • n 2 n 1 .L that there exasts an A:R ->R such that M ={(ax,az); a2eL2, ax=Aa2} and

~1/2 1 P 11(2 C¢tl • 1/2 -/2 T that for generic data e (d,a)= ~ .. with c~:=S a2 and Q:=S (A S~A+S +

t t ]l w¢ II Y y Y y YY T - / 2 . -"/2 -1 " 1/2 . . A Sxy+SyxA)Syy. Define A:=Sxx(A+SxxSxy)~yy , then a direct calculation shows

that Q=I-vETzvT+ATA. So eP(d,M) clearly is minimal on N(L2) if and only if , J / 2 . , . ~ 1 / 2 ,

Ac~=0 for all ~ e ~ y ~ 2 , i.e., (Z_~yy)[L2=0. As Sxx>O it follows that

AlL2=(-S;lxSxy)lL2which corresponds to 31=M*(L2). From the last line of the P * //2

proof of proposit ion 2-18 it follows that e (d,31 (L2))=E(I_V.~TsvT)1/2(SyyL2)=

P r o o f o f c o r o l l a r y 2 - 2 0

T 1/2 3- Let MeN with cP(M)<_r~2-k,, hence dim(M~)>k, and let L2:=V SyyM 2. It follows from

P P * 3- lemma 2-19 and proposit ion 2-10(i) that for generic data e (d,M)>e (d,M (312))=

~(I_sT~)I/2(L2)>_S(I_zT~)I/2(L~.), where L*k=span{cl,...,% } a s ([-ETE) 1/2 has _ , . 2 ,1/2

eigenvalues Al>_...>__),n2>__O with ai=~x-an2_i+l) , ie[1,n], and as %2_i+ 1 is an

eigenvector corresponding to ),~. v:TeV~ ~t-L_r * Hence minimal misfit is achieved by the model M*tM2) w i t h A f f ~ y ylv~l[ 2 - - ~ k o

2 6 8 Appendix

± (i} direct calculation then shows that M2=span{a 2 ; ie[1,k]} and that

, ~ -1 ,~ ( i ) ( i ) 31*'31±'=-x,Y'eR'q×R"2; ~ 2 ) { ~ ) -* <-SxxSxya2~x>+<a2,y>=O for all - - D x x ~ x y a 2 = - f f i a , ~ SO ± n 1 n 2 • . . * a2~M2}={(x,y)eR ×R ; <a~'),y>=a~.<a~*),x>, ~[1,k]}----mk. Moreover lemma

P * * 21/2 2 1 / 2 2-19 implies that e (d,mk)=e(x_~Tz)l/2(Lk)=((1-ak) , . . . , (I-a,) , 0,...,0). •

P r o o f of p ropos i t ion 2-21

According to lemma 2-19 it suffices to determine those subspaces M~ecR n2 for T I/~ ± which.e(i_xTs)V 2 (L2) is minimal, where La:=V SyyM2~ and to accept the models

P . . 31"(312). The requirement e (3l)<cto ! is eqmvalent to dim(L2)= P n2-c (M)>n2-cto~=:k. Let AI>..._>An2>_O denote the singular values of (I-ETE) l/2,

A "1 2 .1/2 i.e., i=(-0"n2_i+1) ~ so especially for r<n 2 A 1 . . . . . An2_r=l and for r*>O

Anz_r*+l . . . . . An2=O. In the notation of section 2.1.2 L~=span{ex,...ek} , and * - 1 / 2 / . . *

in the proof of corollary 2-20 it was shown that 3I (Syyl Lk)=M~, P • 2 1 / 2 el(d,Mk)=(1-aD •

(i) If ctol<n2-r then k>r and hence An2_k=An2_~+,=l. According to

proposition 2-10(i) and (iii) the optimal models are obtained by taking

L2=L'+L" with L'=L~ and L"cspan{ei; ie[r+l,n2] } with dim(L")=n2-Ctol-r , i.e., * . ±

L2DL r and dim(L2)=n2-cto 1. This is equivalent to dim(M 2)=n2-eta and * -1/2 * * McM (Sy y VL=)=Mr.

P * * P * 2 1/2 (ii) Clearly c (Mr*)=n2-r <cta and e (d,Mr.)=(1-ar*) =0, hence eP(d,31**)=O. P * * So it suffices to prove that {c (M)<n2-r , ~P(d,M)=O}~ {M=Mr* }. If

r -1/2

eP(d,M)=O, then there holds M-Lcker(~SxxSxY])=tsyx syyj [SXoV Sy °1/2y vJ ] " ker( Izx T 7 ) ) =

x~ v _5/2 ] . span{ _~ ; ~[1 , r ]}=span{(al,-a(20); ,~[1,r ]}, where e~ and Sy y v.j

e~' denote the i - th unit vectors in R nl and R n2 respectively. If in addition

ce(M)<_n2-r*, then dim(M'±)=dim(lll¢)>_r *, hence M±=span{(a,(i), _a~,));" i~[1,r*]}

and M=M**.

(iii) I f an2_Ctol>O'n2_etot+, t h e n An2_k>An2_k+ 1 and according to proposition

2-10(ii) we get L2=L ~ with corresponding model bl~ 31" 2 ctot

( i v ) I f O ' n 2 _ C t o l = O ' n 2 _ e t o l + l then An2_k=An2_k+l~ SO according to proposition

2-10(i) and (iii) the optimal models are obtained by taking L2=L'+L"

where L'=span{e,,...,eel } and L"cspan{eq+l,...,ec2 } with dim(L")=k-dim(L')= ne-Ctoz-q. The corresponding models are M*(L2)=M*ac~L where L±cM(a,2_%l) ± with dim(L-L)=dim(L"), so M(an2_Ctot)CL and cP(L)=n2-dim(L")=Ctoz+C,. •

C h a p t e r V 269

P r o o f o f p r o p o s i t i o n 2-22

P . . . . * , ,~ 2 ,1/2 to l (i) Clearly el(a,:ln2)=tl-an2 ) <Q and cP(M~2)=0, hence it suffices to show

that {cP(M)=0, eP(d,M)<_eP(d,3I*2)}..~. {M=M~*2}. This follows from lemma

2-19 as for cP(M)=0 bl~=N "2 and 31"(Rn2)=74"2.

(ii) If cP(M)<n2 then dim(M~)_>l and according to corollary 2-20 e~(d,M)_> (~ 2 , ~/2 to~ ±-al)->¢1 • Hence cP(M)=n2, so M~={O} and hence M'I-={O}, i.e.,

M : ~ n l + n 2 .

• ~ 2 , 1 / 2 t o l , .~ 2 1/2 (iii) Let k:=r and note that (l-ak) <~1 <I=(l-ak+l) • The result then follows

from the proof of (iv). P 2 1/2 t o l (iv) If cP(M)<n2-k then according to corollary 2-20 el(d,M)>_(1-a~.+l ) >Q .

P • . ~ , * , ,~ 2 , V 2 t o l As cP(M*k)=ne-k and el(a,:lk)=(l-ak) <el it follows that the minimal

achievable complexity is n2-k. As ak>ak+~ it follows from proposition

2-21(iii) with cto/:=n2-k that the optimal model is M~.. •

P r o o f o f t h e o r e m 4 -4

Let /~ be as defined in step 5 of the algorithm. Due to assumption 4-3(ii) B is ± ±

uniquely defined. It follows from steps 3.0 and 3.2 that vt(Lt).l_vt(Bt_l+Sl3t_l) and it inductively follows from assumption 4-3(iii) that B~ as defined in

steps 2.2 and 3.2 indeed exactly consists of the t - th order laws claimed by /3

as defined in section II.3.2.4. Then (iv) follows from the definition of L D in

section II.3.2.5 and (ii) is implied by proposition II.3-10.

We prove (i) and (iii) by induction. For t=0 the restriction c(8)<Cto / t o t implies that at least e 0 zero order laws should be accepted, cf. definition

4-1. As for such laws eD(~,r)=[]S1/2(~v,O)vo(r)Tll/llvo(r)]] it follows from

assumption 4-3(ii) and p~oposition 2-10(ii) that the unique optimal solution

is given by V 0 as defined in step 2.2, with corresponding misfit (aq_e~oI+l , ...,aq, O,...,O) where a k is the k- th singular value of Slh(~,0). This shows

(iii) for t=O. Note that due to assumption 4-3(ii) and the lexicographie

ordering of misfits, cf. definitions 3-4 and 3-9, it is suboptimal to accept

more than e t°z laws.

Next suppose that optimality of step 3.2 is shown for steps v with v<t-1

for some t>l. According to definition 4-1 it follows from the requirement

c(B)<cw~ and the fact that err 0/ equations of order v have been accepted,

270 Appendix

T ~ Z t - - 1 , tha t a t least e~ °~ equations of order t have to be accepted. Moreover,

according to definition 3-3 we have to minimize the misfit of the newly 3_ 3- .1_

accepted laws in [ v t ( B t _ l T S l 3 t _ l ) ] . NOW for R~N~×q[s] there holds {vt(r)3. 2- .L v t (Bt_ l+SBt_l )}** {3aeR qt such that vt(r)=aTpt}. Hence for such laws t we get

D ,T J /e , e (~,r)=l{rfol[/lIrll=(vt(r)S(ff),t)vt(r J ) /I]vt(r)ll=l}SVt~aH/lla]}, where St:=PtS(~,t)P T. Here we used the fact that Ptpr=Iqt . Assumption 4-3(ii) and proposition

2-10(ii) imply tha t the unique optimal solution for step t is given by Vt of

step 3.2. The corresponding misfit is (aqt_e~Ot+l,...,aqt,O,..,O) where a k is ¢v2 the k - th singular value of o r , which shows (iii) for step t. This concludes

the inductive proof of (i) and (iii). •

P r o o f o f t h e o r e m 4 - 6

Due to assumption 4-5(ii) B in step 5 of the algorithm is uniquely defined.

Now (ii) and (iv) follow from assumption 4-5(iii) as in the proof of theorem

4-4(i i ) and (iv). We prove (i) and (iii) by induction. For t=O it follows from the ordering

of definition 3-12 that c o should be minimized~ i .e , the number of zero order

laws should be maximized. As for such laws eD(~,T)=[[S1/2(~,O)Vo(r)T[[/[[VO(r)[[, it

follows from proposit ion 2-10(i) and from step 2.1 of the algorithm tha t the D . . . . . tol

requirement e0,1~w,o)<e0 implies that at most e 0 laws can be accepted. Given

the ordering of definition 3-12 the misfit of these e 0 laws should he

minimized. Proposition 2-10(ii) implies that the unique optimal solution is

given by V 0 as defined in step 2.2, with corresponding misfit (aq_eo+l,...,aq,

0,...,0) where a k is the k - th singular value of Sv2(~,0). This shows (iii) for

t=0.

Next suppose that optimality of step 3.2 is shown for steps ~- with T_<t-1

for some t_>l. According to definition 3-3 we have to minimize the misfit of -1 _L 3_ .L

newly accepted laws rev t {[vt(Bt_x+sBt_l) ] }. Now r is in this set if and only

if there exists an aeR qt such that vt(r)=aTpt and then eD(ff~,r)=lls1/:all/llall with

St:=P tS( ffJ,t )P~'.

First assume tha t e~'_<e~. The ordering of definition 3-12 implies that a

maximal number of t - t h order laws should be accepted. Due to the requirement D . . . . . . tol

et , l (W,b)<Q it follows from proposition 2-10(i) and step 3.1 of the

algorithm that a t most e~ t - t h order laws can be accepted. The ordering of

Chapter V 271

definition 3-12 and proposition 2-10(ii) then imply that the unique optimal

solution is given by V t as defined in step 3.2, with corresponding misfit ¢1/2

(aqt_e,t,+l,...~aqt,O,...,O), where a k is the k - t h singular value of ,~t, which

shows (iii) for step t.

Finally assume that e~'>e~. Assumption 4-5(iii) implies that q-e~

independent laws or orders T<t--1 have been accepted. It then follows that at

most e~ independent laws of order t may be accepted. Indeed, otherwise the

resulting set of laws cannot be bilaterally row proper and from proposition

II .3-8 it follows that then there would exist an equivalent set of laws with

more than q-e~ laws of order at most t-1. According to steps 3.1 of the

algorithm for T<t--1 this would necessarily lead to unacceptable misfits.

Assumption 4-5(ii) , the ordering of definition 3-12 and proposition 2-10(ii)

imply that tim unique optimal solution in this case is given by Vt as defined

in step 3.2 with misfit as given in (iii).

This concludes the inductive proof of (i) and ( i i i ) . •

P r o o f o f t h e o r e m 4 - 8

Assumption 4-7(ii) implies that step 3.1 is well-defined, assumption 4-7(iii)

that step 3.2 is well-defined and that /~ in step 5 is uniquely defined. It t ,1.

follows from steps 3.0 and 3.2 that vt(Lt}3-vt(Ft_ls }+vt(Bt_l) and it

inductively follows from assumption 4-7(iv) that B~ as defined in steps 2.2

and 3.2 indeed exact ly consists of the t - t h order laws claimed by g as defined

in section II.3.2.4. Then (iv) follows from the definition of LPt in section

II.3.2.6 and (ii) is implied by proposition II.3-10.

We prove (i) by induction. For t=O the optimality of step 2.2 follows

from theorem 4-4, cf. definition 3-7. Next suppose that optimality of step 3.2

is shown for steps r with v<t-1 for some t>l. According to definition 4-1 it

follows from the requirement c(B)<cto I and the fact that er t°l equations of

order v have been accepted, T<t-1, that at least ett °t equations of order t

have to be accepted. Moreover, according to definition 3-7 we have to minimize t -1. -1.

t h e misfit of the newly accepted laws in [vt(Ft_ls)+vt(Bt_l)] . Now for t .l. reRlt×q[s] there holds {vt(r)-l-[vt(Ft-lS )+vt(B t -1)]} ~=~ ( 3a-eRqt, a+ eRe-hi-1

T r a'- [~ala_ ~t):=Pt.col(~(to+i),...,~(to+i+t)), such that vt(r)=(a_,a+)Pt}. Let . - + ,

i e [0 ,q- t0- t ] , then for such laws r we get eP(~,r)= II r~l[ /II r*~l l t =

272 Appendix

T ~ T ~/2 T ~ P ~(t) (a P~S(w,t)Pta) ]tla+P2twl]t=e (z ,a), where the last, static, predictive

misfit is defined as in definition 2-15 with n~:=q~, n2:=q-ntq ,

x~:=Plt" c°l(w(to+i),"',~v(to+i+t-1)) and Yi:=P2t "W(to+i+t), i~[O,tl-to-t]. Due to the requirement vt(r)lvt(13~t_l) at least ett 0/ independent a+ functionals

need to be accepted. Defining "gtoz:=q-nt_rett 0/, cf. section 2.2.1, we obtain

the optimal relations by applying proposition 2-21(iii), due to assumption

4-7(iii). Note that assumption 4-7(ii) implies that the genericity conditions

in proposition 2-21 are satisfied, cf. section 2.2.3. So proposition 2-21(iii) implies the optimality of step 3.2 in step t of the recursion, cf. the

definition of 3l*[a in section 2.2.3. This concludes the inductive proof

of (i).

Finally the expression for the misfit in (iii) follows from (i), (iv),

and corollary 2-20. •

P r o o f o f t h e o r e m 4-10

Assumption 4-9(ii) implies that step 3.1 is well-defined, assumption 4-9(iii) tha t step 3.2 is well-defined and that t3 in step 5 is uniquely defined. Now

(ii) and (iv) follow fl'om assumption 4-9(iv) as in the proof of theorem

4-8(ii) and (iv). We prove (i) by induction. For t=0 the optimality of step 2.2 follows

from theorem 4-6, cf. definition 3-7. Next suppose that optimality of step 3.2

is shown for steps r with ~-<t-1 for some t_l. According to definitions 3-7 and

3-12 we have to minimize the misfit of a maximal number of newty accepted laws -1 t 3_ 3_ r~v t {[vdF~qs )+vdBt_l)] }. For such laws eP(~,r)=eP(z'4t),a) with ~t) and

a as definied in the proof of theorem 4-8(i). First suppose that in step 3.1 P . . . . . tol

e~'<e~. Due to the requirement ct,flw,o)<¢t if follows from proposition

2-22(iv) and the ordering of definition 3-12 that the unique optimal solution

is given by step 3.2, cf. the proof of theorem 4 - 8 ( i ) . Note that assumption

4-9(//) implies that the genericity conditions in proposition 2-22 are

satisfied, cf. section 2.2.3. Next suppose that in step 3.1 ¢~'>e[. Assumption

4-9(iv) implies that at step t at most e[ independent laws of order t may be

accepted for the reason given in tim proof of theorem 4-6. Assumption

4-9(iii), the ordering of definition 3-12 and proposition 2-22(iv) imply that

the unique optimal solution in this case is given by step 3.2.

Chapter V 273

Finally the expression for the misfit in (iii) follows from (i), (iv)~

and corollary 2-20. •

P r o o f o f p ro po s i t i on 5 - 2

As a simple example take B=(Rq) z. For any web and any :7- of the finite length

there exist B'~B such that wIT~B]T and dim(8')<_q.#(:7"), e.g., B':={w~(Rq)Y-; (a#(:7")-l)w=O}. Hence B~P"(wlS,- ) for any wlv-e(Rq) :T. •

P r o o f o f p rop os i t i on 5 - 3

We give the proof for P~ctol, pD and ~D etot P~tot' as similar arguments hold true

for the other procedures.

First suppose that cto t is given. Let eto~:=e(ctot) be the equation

structure corresponding to Ctot, cf. definition 4-1. If etol=O, then it

follows from definition 3-9 that pD is not consistent for the same reasons c tot

as given for pU in the proof of proposition 5-2. If there is t~:~+ with ctt°t>_l, then /~eB with e~(13)=O cannot be exactly identified, hence pD is not

c to t

consistent. D t o l

Next suppose that Q~ is given and consider P~tot" If et,l_<0 for some

tJ /+ , then for every 8eB (exact) identification is impossible for :7.

sufficiently large, cf. the interpretation following assmnption 4-5. If on the

other hand tot ^ pD ~t,l>u for all te77+, then especially e0,1>ut°t ~ so ~tot will accept

laws of order zero for w]2-e(Rq)T of sufficiently small norm. Not having this

sufficiently small norm clearly is not a generic property for any BeB, as B is

Especially eo(~)---~O , --/ '3

linear, if BeB with then ~tol cannot 8 exactly identify

for generic time series~ h~nce pD is not consistent. v to t

t o l Finally for ~tot note that Q,x<u for some tJ /+ implies that for

every BeB identification is' hnpossible for 3" sufficiently large, that ett°,tl>O for some feZ+ implies that, e.g., (Rq) z cannot be exactly identified for

tot ~ for all t~F+ implies that ~D is not generic time series, and that ¢t,l=U etot consistent for the reasons given in the proof of proposition 5-2. •

274 Appendix

P r o o f o f t h e o r e m 5 - 4

For given BoB c we have to prove that generically in w~B there holds

P~(~)=P~e(~)={B} for ~v=w[gr with #(: / ' ) sufficiently large.

In the proof we use the following lemma. Let [l(Y):=(#(:Y)-q)/(q+l) and let

S(~,t) denote the empirical covariance matrix of order t as defined in

sections 3.3.1 and 4.2.1. Further for Vc(R:×q) t+1 let VT:={ve(g(q)t+:; vTev}.

Lemma 5 - 4 For t<d(9 v) generically in wEB ker(S(w[T,t))--[vt(B~)]T.

P r o o f o f l emma 5 -4

[vt(B-7)]Tcker(S(w[sv,t))''. and hence it suffices to show that gen. in Evidently

we13 dim(ker(S(wlT,t)))<dim(vt(B~)), i.e., rank(S(wlT,t))>_q(t+l)-dim(vt(B~))=

q(t+l)-Zk=o(t+l-k)ek, where {ei; teZ+} is the tightest equation structure of

/3, cf. proposit ion IV.2-2(i) and the remark following definition II.3-9. Let

T:=#(9") and relabel the time instants such that :T=[1,T]. Then S(w]5~,t)= 1 iH(w)H(w)T, where H(w)eR q(t+:)×(T-t) is defined by

T -

w(1) w(2) ... w(T-t) w(2) w(3) ... w(T-t+l)

H(w):= w ( t + l ) w( t+2) ... w(T)

It hence suffices to prove that there exists a web with rank(H(w))_> t * q(t+l)-Zk=o(t+l-k)ck=:rw, as this then also holds true generically on B.

Nota t ion . We use the following notat ion and concepts.

Let /3 have a minimal input /s ta te /output realization BiHo(A,B,C,D), i.e., there exists a permutat ion matrix // such that Ill3={(u,y)~(Rm)Z×(RP)Z;

I;; lI.l Bx~(Rn) z such that a = }, where n is the number of states, m the

number of inputs, and p:=q-m the number of outputs. We refer to definition ~ N N

II.3-24. As BeB c it follows that (A,B,C) is a minimal triple, by which

we mean that (A,B) is controllable, cf. section IV.3.2, and that (A,C) is

observable, cf. the proof of proposition IV.3-5, i.e.,

rank([B AB...A B])=rank(col(C,CA,...,C~4n-1))=n. For given FeR m×p let (A,C):=(A+BFC, C+DFC) and for (u,x,y)eYil, lo(~i,B,C,l) )

Chapter V 275

let v:=u-FCx. Then IIB={(u,y);3(v,x) such that = ~ }. Hcncc

@e/3][1,r] if and only if there exist x0e~ n and v[[1,T]e(Rm) T such that for all

~'e[1,T] l l~(T)=r~(r) l with ~t('r)=FCx(T)+vO- ) and ~(7)=Cx(T)+Dv(r), where Ly(")J

r - I ~ - 1 k - 1 ~ x(v):=A x0+Ek=IA By(r-k), Te[1,T].

We call A~R n×n cyclic if there exists an xo~R n such that

det([x0 Axo...A~-Xxo])#O. For ze(Ra) z let H(z)cR a(t+l)x(T-t) be defined by

) ... z ( T - t ) ] z(3)

L z ( i + l ) z(t+2) ,.. ziT) ]

Finally, for Mi~R nl×n2, i=1,2, let M1~312 denote that 311 can be transformed

into 312 by means of elementary row operations, i.e., {M~~312} : ~ {3SeR ~*'4

nonsingular such that S3I~=Mz}. []

We now show that there exists a web with rank(H(w))>_rw, by choosing

appropr ia te F, x0, and v l[1,T]. The proof is split in the following three

parts.

(i) 3FeR m×p such that A:=~4+BFC is invertible and cyclic;

(ii) for F as in (i) and for w~B with //w= and [ u] = ]FO there holds

L rJ L c r n IH/ /)--rankl whet0 ....

(i~i) there exist x0eR" and V[[I,TIE(Rm) T such that for x(1):=x o and

x(T+l):=Ax(v)+Bv(T), T~[1,T-t-1], there holds rank([H~vl])>rw. l_ .l

The desired result then follows from (ii) and (iii).

(i) We have to prove that there is an FeR "*×P such that A:=A+BFC is (1) cyclic

and (2) invertible. As these are algebraic conditions in F, cf Kailath

[33, 1emma 7.1-2], it suffices to prove that (1) and (2) can be satisfied

individually, as this even implies that conditions (1) and (2) are

simultaneously satisfied for generic F.

For (1) we refer to Kailath [33, lemma 7.1-2 and section 7.2.3].

Next we consider (2). As (A,B,C) is a minimal triple it follows from

276 Appendix

Kailath [33, lemma 7.1-1] that there exist tieR" and yeR p such that for b:=Bfl

and C:=TTc also (~4,b,c) is a minimal triple. Let a basis in R n be chosen such

that {A,b) is in control canonical form, cf. Kalman, Falb and Arbib [39, 11 1 1 - 1 J:

section 2.4], i.e., if det(sI-A)=s +Ek=Oaka and a:=-(ao,ai,...,a~_1) , then in

this basis ~=[0 In-l~a j '~ b=(o,...,O,1) T, and c=(q,...,Cn) for some cieR,

iE[1,n]. If a0¢0 then A is invertible, hence F=O satisfies (2). If ao=O then

observability of (A,c) implies that c1~0 , hence for F:=fl7 T there holds A= (o In_i) , j with a':=(c,,c2-a,,...,cn-an_,) , and as ci~0 A is invertible.

I°i] Ii ]I:l E:I (ii) Let F be as in (i), (A,C):=(A+BFC,C+DFC), = ~ and w:=ll . Then

by means of elementary row operations it follows that H(.w)~]u(~)]= L"(ylJ

~ H(u) ] _ [ll(FOx)+H(v)]- [ H(v)] H(Cx) +H(bu)J L n(~x) J k**(~.)j"

Noting that for I<r,<_T2<_T there holds X(T2)=ArZ-rlX(71)+ ~ r 2 - r 1 , k - l ~ , , . , -k=l A tsv(rz-~), it follows from the Hankel structure of H(v) and H(Cx)

[ "(~)] _ that LH(~x) J ~---[H(')] ~ where M:=col(C,CA,...,CAt).[x(1)x(2)...x(T-t)].~~ For

example, x(v+l)=Ax(r)+Bv(r), r~[1,T-t], hence by subtracting CB times the

matrix consisting of the first m rows of H(v) from the matrix consisting of

rows p+l,p+2,...,2p of H(Cx) this latter matrix is transformed into

CA[x(1) x(2)...x(T-t)], and similarly for the other rows.

follows that H(w)-H(v) As col(C,6"A,...,CAt)-col(C,C.Ti,...,C~i t) it [M ] with f - __ . "1

~'/:=eol(C,C,4 .... ,C.4').[x(1) x(2)...x(T-t)], hence rank(H(w))=rank(l '~=)/ ) L*~ 3

(iii) Finally we prove that there exist XoeR n and v I[1,T]e(R') T such

that for x{1):=x o and x(r+l):=Ax{r)+Bv(r), re[1,T-t-1], there holds

rank( [H!v)] )>_r~. M ~ t *

Let rx:=p(t+l)-Sk=o(t+l-k)%=rw-m{t+l ). Denoting the i - th component of v by vi, ic[1,m], we define v l[1,r]~(R') T

by vm_~(t+rx+k(t+l)+l):=l , k~[0,m-1], and 0 elsewhere. Note that

t+rx+(m-1)(t+l)+l=rw<_q(t+l)<T-t , as t<[l(:T)=(T-q)/(q+l). It is a simple matter

of explicitly writing out H(v) to conclude that rank(H(v))=m(t+l) and that in

H(v) the first r x columns are zero. As v t[t,r~]=O there holds that T - 1 x(r)=A x0, re[1,rx]. Now suppose that there is an XoeR n such that

rank(col(C,C~i,...,C~4t).[xo Axo...ArX-*Xo])=rx. Then the matrix consisting of

C h a p t e r V 277

the first r~ columns of /~ has rank rx, and it easily follows that the first

columns of |n~v)] are linearly independent, hence rx+m( t+l )=rw rank([n(v)l)>_rw, as desired.

L ~ ]

I t remains to prove that there is an x0eR ~ such

ttlat ~lo:=col(C,C.74,...,C~4t).[xo Axo...Arx-lXo] has rank r x. It follows from

Willems [73, table 2 and theorem 6(vi)] that rx<n and that . . . . . t t k * t * rank(col(C, CA,...,CA ))=Sk=o(P--Sj=oej)=p(t+l)--~k=o(t+l--k)ek=r x. Now suppose

that rank(~/o)<rx for all x0eR n. Then there would exist a k<r x such that for

generic xocR" the (k+ l ) - th column of ~I0 is linearly dependent on the foregoing n - 1 - k .

ones, say for xo~VcR n where V is generic• Then V:-- zNo.= A V : = { x ~ ; A~x~V for

all i~[O,n-l-k]} is also generic, as A is invertible. For every

xocV rank(col((~,C~i,...,~{t).[xo Axo...A"-lXo] )<_k<rx. However, as A is

cyclic, for generic xo det([x 0 Axo...An-tXo])#O and hence generically on

R n rank (col (C, CA, ...,c,4t).[Xo Axo...An-lxo])=rank(col(C,C~t,...,C'~tt))=rx. This

contradiction shows that there is an x0eR '~ with rank(M0)=rx, as desired.

This concludes the proof of lemma 5-4. •

D D Using this lemma we now first prove the theorem for POe. Let Vt:=vt(Lt) be tile

descriptive complementary spaces of B as defined in section II.3.2.5 and let

nt:=dun(Vt)=et(B), see proposition II.3-10. Moreover let t :=max{t;n#=O} and

let {t; nt#O}={tx,...,tc} with O<t<t2<...<tc_l<tc:=t*. We now show that gen. in

web /~e(wl:?-)={B } for every T with [I(T)>t*, i.e., #(T)>t*(q+l)+q.

Let :7- be such that ~(:T)>t*. Now P~ only accepts laws with descriptive n l x ~ ' r , misfit zero. Note that for t_<~[(T) and r ~ t is] there holds {eO(~,r)=0}¢=~

{vt(r)~ker(S(~,t)) }. I t hence follows from lemma 5-4 that gem on 13 pD

accepts no law of order t<tl, as then vt(Blt)={0}. For step t=t 1 gen. on 13

ker(S(w[:T,t))=vD 1 and hence gen. /~e exact ly identifies these laws on step t t.

Now suppose that for some k~[2,c] PO e gen. on steps tf identifies V D t i ,

i~[1,k-1], and gen. no laws are accepted on steps t_<tk_ I with t 6 { t l , . . . , t k _ l } .

- ± ~± ~± LD={0), and it For steps t~[tk_l+l,tk-1 ] there holds that l~ t =~,_l+sDt_l, as

follows by induction from lemma 5-4 that gem on /3 the rows of Pt as defined

in step 3.0 of the algorithm of section ,1.2.2 span [vt(B~)] ± and that gem on

13 ker(PtS(w }T,t)FT)={O}. So gen. no laws are accepted for steps ± ± . £

t~[tk_t+l,t~-l], and gen. the rows of Ptk span [Vtk(Btk_l+sBtk_1) ] . Lemma 5-4

2 7 8 A p p e n d i x

and the algorithm of section 4.2.2 imply that hence gen. on /3 for step t k P~e _1_ .L .L _L V D exactly identifies Vtk(/3tQn[Vtk(Btk_l+S/3tk_l) ] = tk. It follows by induction

that on step d(Y') P~e gen. has identified exactly vDi, ti<_t*, i.e.,

/~ee(W[:T)={/3 }. Note that the number of steps is finite and that a finite

intersection of generic sets is generic.

This proves consistency of P~e. The consistency of P~e follows from this

result. Note that for order zero the procedures coincide, while for steps t_>l

{eP(~,r)=O}.~{eD(~o,r)=O), provided that [[r*~][t~0 where r* is the leading

coefficient vector of r, cf. definition 3-5. As always on step 0 the set of P D zero order laws Vo=V o is identified it follows that gen. on t3 in all steps

* * P l_<t_<d(T) I[r*~]lt~0, as it is required that r ±Ft_l, hence r ±V 0. The

consistency of P~e then in]plies the consistency of P~e. •

P roo f of theorem 5-5

Lemma 5-4 and the proof of theorem 5-4 show the following. We recall that for

given interval of observation :7" the procedures P~e and P~e only consider the

model class ~(:T) for identification, cf. definitions 3-8, 3-13 and 3-14.

Lemma 5-5 Generically on 8eBc, P~(w[5~)=P~(wl:T)=/3(Y-):={w'c(Rq)z; r(a)w'=O for all reB~ where t:=d(:T)}.

_l_ .L pD and /~e evidently are exact. Moreover, as B tc/3t+l it follows that

B([to,tl+l])=B([to-l,tl])cB([to,tl] ) and lemma 5-5 implies that pD and P~ are

bilaterally monotone, i.e., the identified model becomes more strict if more

observations are available. To prove linearity, let B1,/3eeBc, then /~lq-/32e~]c~ [ 1 ± ,t3i) t ~(/31+B2) t , i=1,2, hence (/31+B2)(T)~Ba(Y')+I3a(:T) and lemma 5-5 implies that

P~e and P~e axe linear. That these procedures axe truthful is evident from lemma

5-5. Finally, for given 7 the procedures P~ and P~ only identify models in the

class •(:T), cf. definition 3-8, and according to lemma 5-5 models in B(:T) are

strongly corroborable by P~e and P~e, which proves that these procedures are

strongly prudential. •

C h a p t e r V 279

P r o o f o f t h e o r e m 5-10

We first s ta te and prove two lemmas which will be used to prove the theorem.

Nota t ion . For A=ATeR n×n let a(A):=(al,...,an) , with al>...>a,, denote the

ordered set of eigenvalues of A. Let A=U•U T, ~=diag(al,...,an) , uuT----uTu~-Im

and for k<l with ak_l¢ak, al~=al+l, let /4[k,1]cR" denote the space spanned by

the columns k,k+l, . . , l of U. Consider the collection of finite dimensional

linear spaces £:={L; 3hen with LeR n, and L linear}. A sequence (Lk; keN) is

defined to converge in the Grassmannian topology if there is an LeL:, Lc~ n,

such that for k sufficiently large LkcR', dim(Lk)=dim(L), and such that there

exist choices of bases in L k which converge in Euclidean sense to a basis of

L. We denote this by Lk (a~ L. The sequence is defined to converge in the gap

topology if there is an LeE, LcR", such that LkCR n for k sufficiently large

and such that g(Lk~L)-~O for k-~c% where g denotes the gap between L and Lk, cf.

definition IV.5-1 and Kato [40, section IV.2.1]~ or Stewart [66~ section

2.1]. []

Lemma 5-10-1 The Grassmannian and gap topologies are equivalent.

Lemma 5-10-2 (i) The mapping a:A-~a(A) is continuous; (ii) if in a(Ao)

for some k<l ak_l~a k and az~al+l, then the mapping A-~ ' [k j l is

continuous in A0, in the gap topology; (iii) if Ao=ATo>O then the mapping

A-~ A -1/2 is continuous in A 0.

P r o o f o f l emma 5 - 1 0 - 1

Let {Lk; keN} be given. Note that convergence in the Grassmannian or gap

topology both imply that there is an n~N such that LcR n and LkCR n with

dim(Lk)=dim(L)=:d for k sufficiently large, cf. Stewart [66, section 2.1].

First suppose that Lk (a~L. Then there are matrices B and B k of full

column rank with Lk=im(Bk) , L=im(B), and ]lBk-BI]->0 if k-~o0. The orthogonal

projection operators P and Pk on L and L k respectively are given by T -1 T P=B(BTB)-IB T and Pk=Bk(BkBk) Bk, hence IIPk-P[]->o, and Stewart [66, theorem

2.2] implies that g(Lk,L)-->O for k-)oo, cf. lemma IV.5-2.

280 Appendix

Next suppose tha t g(L~,L),O. Let L=im(B) where B has full column r ank d.

For all e>0 there is a K E such tha t fo r k>K e the re exist {b~(k,e),...,

bd(k,e)}cL ~ such tha t fo r B(k,s):=(b~(k,e),...,bd(k,e)) I[B-B(k,e)ll<e, cf.

def ini t ion IV.5-1. For e suff ic ient ly small B(k,e) has full column rank and

hence fo r k suff ic ient ly l a rge the columns of B(k~e) fo rm a basis of Lk, i.e.,

L~(a~L. •

P r o o f o f l e m m a 5 - 1 0 - 2

(i) We r e f e r to S tewar t [67, co ro l l a ry 6.5.11].

(ii) Let A=U~U T with uuT=uTu---In, ~=diag(al,...,an) , al>_...>an, and let k<l be

such tha t ak_l~a k and at~aZ+l. Let UI~R n×(l-~+l) consist o f columns

k,...,l of U and U2eR n×(~-l+k-1) of the remaining columns of U, so

//[k,tl=im(U1). Let ~:l:=diag{ ak,. . . ,al ), Z2:=diag(al,...,ak_1,at+l,...,an), so

A=(U 1 U2)I~ ' ° l ( U l U2) T. Then the so -ca l l ed sepa ra t ion of Z 1 and £2 is ku 23

min{lajl-aj2[; jle[k,l], j2~[1,k-1]u[l+l,n]}>O. Now suppose t ha t Ai=AT->A. ( i ) _ ( i ) ( i ) ( i )

Let A i have e igenvalues a~ >...>__an, then according to (i) a j >a k for

j<k and @i)<ffli) fo r j>l if i is suff ic ient ly large. Let Ai=

~(20 (U~ i) U(20) r be a decomposi t ion of Ai ana logous to the . /Tl(i)~ 71(i)

one of A, so especial ly - , ~ t J=~[k,ti. I t follows f rom Stewart [66, section

4.7 and coro l l a ry 2.6], t ha t hn(U~iI).~im(U1) in the gap t opo logy if i-~ce, i.e.,

Ul~!zi->//[k,z] if i->ee. We also r e fe r to Davis and g a h a n [10, sect ion 2 (the

sin20 theorem) and theorem 8.2].

(iii) For A0>0 the mapping A->A -1 is cont inuous in A0, so it suffices to show

tha t A->A ~/~ is cont inuous in A a. I f A=U~UT>O with UUT=UrU=ln and

E=diag(a~, . . . , an) then A~/2=UN~/2U T. According to (i) the mapping A - ~ v is

cont inuous, and by apply ing (ii) and lemma 5-10-1 fo r the e igenspaces of A 0

cor responding to distinct e igenvalues it follows tha t the re exists a choice

fo r U such tha t A.->U is cont inuous in Ao. Hence A , A ~/~ is cont inuous in A 0. •

The two fo rego ing lemmas a re used to p rove the theorem.

N o t a t i o n . Let ~=w~[:T fo r a rea l iza t ion w r of a s tochast ic process w in tile

class G cctot or O(etot,-a) and let APtol(W)=Bc~]~ and APtot(W)=13~e~]~ be as defined C e / .E ~ C

in defini t ion 5-7, cf. p ropos i t ion 5-9. Let Vt:=vt(Lt) and I t:=vt(Lt), where L t

C h a p t e r V 2 8 1

and L t are the predictive spaces in (CPF) corresponding to B c and B~

respectively, a s defined in section II.3.2.6. Let e(Bc) and e(Be) denote the

tightest equation structures of B c and B~ respectively. We will show that for

sufficiently large, a . s . P~ctol(~)::Bc(Y') and P~(~tol,~)(VJ)=:Be(Y') are T:=#(T) singletons. We denote the corresponding predictive spaces by V~(T) and V~(T) and the corresponding tightest equation structures by ec(T) and ee(Y ") respectively. Further we use the notat ion of the algorithms in sections 4.3.1

and 4.3.2 and we use a • to indicate symbols corresponding to these algorithms

when applied to the process w, cf. proposition 5-9. Finally the condition that

# ( T ) should be sufficiently large is denoted by T->oo. []

We now prove the theorem by induction and first consider l °P c lol"

Let weC~ccta. We will prove by induction that a.s. co(T) ~ e(Bc) and

Vt(T ) ~ Vet in the Grassmannian topology. By choosing bases 31(t)(T) in Vet(T) which converge to bases M (t) of V~ and defining R(T):=col(v-tl(31(t)(T)); tcZ+) and R:=col(vtl(M(t)); t~Z+) we get Be(T)=8(R(T)), /~e=B(R), while R(U) -> R in

Euclidean sense where R is bilaterally row proper, cf. proposition II.3-8,

definition I1.3-15 and assumption 4-7(iv). Hence a.s. Be(T)--> Be, cf. section

5.3.2, which shows consistency of P~cto z. I t remains to show that a.s. ee(T)--> e(Bc) and Vct(T)(a~VCL. Consider the

algorithm of section 4.3.1. Note that Ctoa is sensible for T-~ o0, i.e.,

assumption 4-7(i) is then satisfied. As weG~ctoacG it follows from assumption

5-6 that a.s. S(~,t)--> * S(w,t)=:S (t) for all teZ+ if T-~c¢ . So assume

henceforth that w r satisfies assumption 5-6(ii), then it remains to show that

{S(~, t ) -> 5"*(t); teZ+}=~{eC(T)-> e(Bc) and V~(Y)(e~V~ for all t~Z+ if T-> 00}.

First consider step 0 of the algorithm of section 4.3.1. As w~G c it tol

... .(0 .(0 follows from definition 5-8 and assumption 4-7(zzz) that ~r._etOZ>a._etOl+1.

~[ 0 ~ 0

As S(~,0) -> S*(0) it follows from lemma 5-10-2(i) that assumption 4-7(/ii) for

t=O is satisfied for T-> cv. I t then follows from step 2.2 and lemma 5-10-2(ii)

that e~(T)--> eo(Bc) and g(Vo(T),Vo)--> 0, and from lemma 5-10-1 that hence

V~o(Y)(C~V~. It follows from Stewart [66, theorem 2.2] (cf. lemma IV.5-2), that

tlle projection opera tors PI(T) and P~ of step 3.0 for t=l satisfy

IfPI(T)-P~]I -> O. Note that the dimensions of PI(T) are equal to those of P1 if

eo(Y)=eo(8c) , i.e., for T -> c¢.

Next suppose that for some t<max{T; eT(Bc)¢O}=:t* it is proved that for

2 8 2 A p p e n d i x

all k~[O,t-1] ~(T) * ek(B~l, V~(Y)~"~V~, and Pk÷,(7) "~ P~+,. As

S(~,t) S*(t) there holds Pt(T)S('~,t)Pt(T) T • . ,T ->PtS (t)Pt • I t follows from

definition 5-8 and assumption 4-7(ii) for w that S (*) and S (0 as defined in

step 3.1 for ~ have full rank if T-> oo. Let Ct(Y):=(S (t) "1/2S(0 S (t) -~/~ ) _+( + ) and let

C~ be defined analogously in terms of S*. It follows from lemma 5-10-2(iii)

that Ct (7 ) -~ C~. By considering Ct(:T)cT(u) it follows from lemma 5-10-2(i) and

assumption 4-7(ii i) for w that this assumption also is satisfied for ~ if

T-> oo, hence e~(iT)-> et(13c) for T-> oo. Applying lenmaa 5-10-2(i) and (ii) to

Ct(7)cT(7) and cT(y)Ct(7) it follows that in step 3.2 g(VCt(7),V~)-> 0 and

hence Vt(:T)(a~vct if T-> oo. Moreover assmnption 4-7(iv) for ~ is implied by

that for w if T-> oo and hence the laws identified at step t for ~ then are

real ly t - t h order laws, i.e., V~(:T) then remains unchanged for k<t-1. As now

g(V~(:7"),V~.)-> 0 for all k<_t if T-> 0o it follows that in step 3.0 for t+l

[IPt+l(f')-Pt+l]l--> O, cf. lemma IV.5-2. This concludes the inductive part and G c

shows that for t<t* eel(iT) --> el(Be) and Vt(:T) ( ~Y t if T -> 0o.

Finally consider orders t>t*. As et(13c)=O for t>t*, the fact that

et(T)=et(13c) for t<t* and T-> 0¢ implies that for @ it is, for the given eta ,

allowable not to accept any law of order t>t*, for T -> c¢. Moreover, for t<d(7")

A:={Coe(Rq):T;det(S(ffl,t))=O} is a proper algebraic var ie ty and hence has

Lebesgue measure zero, cf. Federer [12, section 2.6.5]. As w ~ G C the

continuity of w implies that S(@,t)>0 a.s. Then also PtS(Co,t)pTt>O a.s., and

definition 3-9 implies that hence et(:T)=O a.s. for all te[t*+l,d(iT)]. This

shows tha t Vt(:T)(G)Vt, tear+, if T -> c¢, as desired.

The consistency of P~(etov~) on G(.tov~ ) is proved in a completely

analogous way by using definition 5-8, the first remark in section 5.3.5, • (0 ) ,-to~,2 assumption 4-9, and the algorithm of section 4.3.2. Note that aq_%>t~ 0 ) ,

(0 ) , - t o / . 2 which implies that for ~ also aq_%>teo ) a.s. for T-~ c¢, and in this case

%(Y-)=%(B~). Similar arguments hold true for steps t<_t*. That Vt(Y-)(a~Vt a.s.

for t<_t* follows in a way analogous to the proof of Vt(:T)(GIVt . Further note

that et(Y')=et(B~)=O for all t>~. Finally we consider te[t*+l,d]. Definitions . , * ( t ) , 2 ,-to~,2

3-12 and 5-8 and the fact that et(Bs)=O imply that l-(a~ ) >tel ) , and it . , ( t ) . 2 . - t o t . 2

follows by induction that hence l - ta~ ~ >tet J and e~(Y-)=O a.s. for all

te[t*+l,d] if T-> 00. It follows that V~(T)(a~v~ a.s. for all re71+ if T-> co, as

desired. Note that now continuity of w is not required.

C h a p t e r V 2 8 3

This concludes the proof of theorem 5-10. •

P r o o f o f t h e o r e m 5-11

Let ~o~Y2~to I and Bo:=P~%ol(~0) , t*:=max{r; e,(B0)¢0 }. Let Ok-* wa if k-~ co,

hence S(Ok,t)-> S(~o,t) for all t<d(Y') if k -> co. In the proof of theorem 5-10

let ~0,~k, S(~o,t) and S(~k,t) play the role of w, ~=wr[y-, S*(t) and S(5,t) respectively. As for w0 assumption 4-7(ii), (iii) and (iv) are satisfied it

follows directly from the proof of theorem 5-10 that these assumptions also

are satisfied for ~k if k-~ co, for all t<t*. Definition 3-9 implies that * ~ * T PtS(wo,t)Pt >0 for all te[t*+l,d(Y')]. From this it follows by induction

that PtS(~k,t)pT>o if k -~ co. In this case et(P~ctol(Cok))=et(13Ü) for all t<d(Y')

and ~ke~CCto I if k -> co. This shows that ~cto~ is open. The proof of theorem

5-10 for t<t* moreover implies that P~cto~(~k).->l~¢to~(ff~o) if k-> co, which

proves the continuity of P~%oz on ~ctol. That J~toz is open and that on this set I~eto I and ~to~ are continuous

also follows directly along the lines of the proof of theorem 5-10 by taking

d:=~(Y'). Finally note that the assumptions on ~etol imply that none of the _tol y2P • restrictions s t is critical. Hence P~%o=~to~ on ~to?

P r o o f o f t h e o r e m 5-12

ff~oe~ctot , then P~ctoZ(50) is a singleton, see theorem 4-4(i) . Let 5k -> 50 Let

if k -> co, hence S(~ok,t ) --> S(Coo,t ) for all t<_gi(T) if k -> co. From the definition

of Y2 D it follows along the lines of the proof of theorems 5-10 and 5-11 c to l

that S(Cok,t ) also satisfies assumption 4-3(ii) and (iii) for all t<~(Y') if

k-> co, and hence that ~C,ot is open. That P~ctol(ff~k)--> P~c,ol(~o) if k-> co also

follows directly along the lines of the proof of theorem 5-10. D D _=D

The results for J-2%oi, P~toz' and t~to z follow in a completely similar

_tol ~D it follows that way. As none of the restrictions ¢t is critical on %ol N D

~ • . ~ . . F - ' t o l ( f f J / = t " = t o z ( ~ for we£2%o F •

REFERENCES

[1] Akaike, H., Canonical correlation analysis of time series and the use of an information criterion, in Mehra, R.K., and D.G. Lainiotis (eds.), System identification: advances and case studies, Academic Press, New York, 1976, pp. 27-96.

[2] Akhiezer, N.I., and I.M. Glazman, Theory of linear operators in Hilbert space, volume I, Frederick Ungar, New York, 1961.

[3] Anderson, B.D.O., and J.B. Moore, Optimal filtering, Prentice-Hall, Englewood Cliffs, New Jersey, 1979.

[4] Anderson, T.W., The statistical analysis of time series, Wiley, New York, 1971.

[5] Box, G.E.P., and G.M. Jenkins, Time series analysis, forecasting and control, ttolden-Day, San Francisco, 1970.

[6] Brillinger, D.R., Time series analysis, data analysis and theory, ltolt, Rinehart and Winston, New York, 1975.

[7] Caines, P.E., On the scientific method and the foundations of system identification, in Byrnes, C.I., and A. Lindquist (eds.), Modelling, identification and robust control, North-Holland, Amsterdam, 1986, pp. 563-580.

[8] Chen, C.T., Linear syste~n theory and design, Holt, Rinehart and Winston, New York, 1984.

[9] Corr~a, G.O., and K. Glover, Pseudo-canonical forms, identifiable parametrizations and simple parameter estimation for linear multivariable systems, Automatica 20 (4), 1984, pp. 429-452.

[10] Davis, C., and W.M. Kahan, The rotation of eigenvectors by a perturbation III, SIAM journal on numerical analysis 7 (1), 1970, pp. 1-46.

[11] Davis, M.H.A., and R.B. Vinter, Stochastic modelling and control, Chapman and Hall, London, 1985.

[12] Federer, 1t., Geometric measure theory, Springer, Berlin, 1969. [13] Finesso, L., and G. Picci, Linear statistical models and stochastic

realization theory, in Lecture notes in control and information sciences 62 part 1, Springer, Berlin, 1984, pp. 445-470.

[14] Fomby, T.B., R.C. Hill and S.R. Johnson, Advanced econo~netric methods, Springer, New York, 1984.

[15] Gantmacher, F.R., The theory of matrices, volume I, Chelsea, New York, 1959.

[16] Gevers, M., and V. Wertz, Uniquely identifiable state-space and ARMA parametrizations for multivariable linear systems, AutoTnatica 20 (3), 1984, pp. 333-347.

[17] Glover, g., All optimal llankel norm approximations of linear multivariable systems and their L ~ error bounds, International journal of control 39 (6), 1984, pp. 1115-1193.

[18] Glover, K., and J.C. Willems, Parmnetrizations of linear dynamical systems, canonical forms and idcntifiability, IEEE transactions on automatic control AC-19 (6), 1974, pp. 640-6.t6.

286 References

[19] Golub, G.H., and C.F. Van Loan, An analysis of the total least squares problem, SIAM journal on numerical analysis 17 (6), 1980, pp. 883-893.

[20] Golub, G.H., and C.F. Van Loan, Matrix computations, Johns Hopkins University Press, Baltimore, 1983.

[21] Guidorzi, R., Invariants and canonical forms for systems structural and parametric identification, Automatica 17 (1), 1981, pp. 117-133.

[22] Hannan, E.J., Multiple time series, Wiley, New York, 1970. [23] Hannan, E.J., and M. Deistler, The statistical theory of linear systems,

Wiley, New York, 1988. [24] Hannan, E.J., and L. Kavalieris, Multivariate linear time series models,

Advances in applied probability 16 (3), 1984, pp. 492-561. [25] Hazewinkel, M., and R.E. Kalman, On invariants, canonical forms and

moduli for linear, constant, finite dimensional, dynamical systems, in Lecture notes in economics and mathematical systems 131, Springer, Berlin, 1976, pp. 48-60.

[26] Heij, C., Exact modelling of a finite data sequence, Proceedings of 25th IEEE conference on decision and control, Athens, 1986, pp. 1743-1744.

[27] Heij, C., Approximate modelling of deterministic systems, in Curtain, R.F. (ed.), Modelling, robustness and sensitivity reduction in control systems, NATO ASI series, Springer, Berlin, 1987, pp. 271-283.

[28] Heij, C., Exact modelling of a finite time series, SIA3I journal on control and optimization 26 (1), 1988, pp. 83-111.

[29] Heij, C., and J.C. Willems, Consistency analysis of approximate modelling procedures, in Byrnes, C.I., C.F. Martin and R.E. Saeks (cds), Linear circuits, systems and signal processing : theory and application, North-Holland, Amsterdam, 1988, pp. 445-456.

[30] Heij, C., and J.C. Willems, A deterministic approach to approximate modelling, in Willems, J.C. (ed.), From data to model, Springer, Heidelberg. To appear.

[31] Hinrichsen, D., and J.C. Willems (eds.), Special issue on parametrization problems, IMA journal of mathematical control and information 3 (2&3), 1986, pp. 59-254.

[32] Jayant, N.S., and P. Noll, Digital coding of wavefo~ns, Prentice-Hall, Englewood Cliffs, New Jersey, 1984.

[33] Kailath, T., Linear systems, Prentice-Hall, Englewood Cliffs, New Jersey, 1980.

[34] Kalman, R.E., On minimal partial realizations of a linear input/output map, in Kalman, R.E., and N. De Claris (eds.), Aspects of network and system theory, Holt, Rinehart and Winston, New York, 1971, pp. 385-407.

[35] Kalman, R.E., Identification from real data, in Hazewiukel, M., and A.H.G. Rinnooy Kan (eds.), Current developments in the interface: economics, econometrics, mathematics, Reidel, Dordrecht, 1982, pp. 161-196.

[36] Kalman, R.E., System identification from noisy data, in Bednarek, A.R., and L. Cesari (eds.), Dynamical systems II, Academic Press, New York, 1982, pp. 135-164.

[37] Kaiman, R.E. , Identifiability and modelling in econometrics, in Krishnaiah, P.R. (ed.), Developments in statistics, volume 4, Academic Press, New York, 1983, pp. 97-136.

[38] Kalman, R.E., and R.S. Bucy, New results in linear filtering and prediction theory, Transactions ASME, Journal of basic engineering, series 83D, 1961, pp. 95-108.

References 287

[39] Kalman, R.E., P.L. Falb and M.A. Arbib, Topics in mathematical system theory, McGraw-Hill, New York, 1969.

[40] Kato, T., Perturbation theory for linear operators, Springer, Berlin, 1966.

[41] Kendall, M.G., and A. Stuart, The advanced theory of statistics, 3 volumes, Griffin, London, 1958, 1961, 1966.

[42] Koopmans,T.C. (ed.), Statistical inference in dynamic economic models, Cowles commission for research in economics, monograph no. 10, Wiley, New York, 1950.

[43] Kullback, S., Information theory and statistics, Wiley, New York, 1959. [44] Kumar, P.R., and P. Varaiya, Stochastic systems: estimation,

identification and adaptive control, Prentice-Hall, Englewood Cliffs, New Jersey, 1986.

[45] Lax, P.O., and R.S. Phillips, Scattering theory, Academic Press, New York, 1967.

[46] Lindquist, A., and M. Pavon, On the structure of state-space models for discrete-time stochastic vector processes, IEEE transactions on automatic control AC-29 (5), 1984, pp. 418-432.

[47] Lindquist, A., and C. Picci, Realization theory for multivariate stationary gaussian processes, SIAbl journal on control and optimization 23 (6), 1985, pp. 809-857.

[48] Ljung, L., Convergence analysis of parametric identification methods, IEEE transactions on automatic control AC-23 (5), 1978, pp. 770-783.

[49] Ljung, L., A non-probabilistic framework for signal spectra, Proceedings of 24th IEEE conference on decision and control, Fort Lauderdale, 1985, pp. 1056-1060.

[50] Ljung, L., System identification: theory for the user, Prentice-Hall, Englewood Cliffs, New Jersey, 1987.

[51] Ljung, L., and P.E. Caines, Asymptotic normality of prediction error estimators for approximate system models, Stochastics 3 (1), 1979, pp. 29-46.

[52] Ljung, L., and T. S6derstr6m, Theory and practice of recursive identification, MIT Press, Cambridge, Mass., 1983.

[53] Malinvaud, E., Statistical methods of econometrics, North-Holland, Amsterdam, 1970.

[54] Moore, B.C. , Principal component analysis in linear systems: controllability, observability, and model reduction, IEEE transactions on automatic control AC-26 (1), 1981, pp. 17-32.

[55] Nieuwenhuis, J.W., and J.C. Willems, Continuity of dynamical systems: a system theoretic approach, Mathematics of control, signals, and systems 1(2), 1988, pp. 147-165.

[56] Northcott, D.G., Lessons on rings, modules and multiplicities, Cambridge University Press, London, 1968.

[57] Payne, H.J., and L.M. Silverman, On the discrete time algebraic Riccati equation, IEEE transactions on automatic control AC-18 (3), 1973, pp. 226-234.

[58] Pernebo, L., and L.M. Silverman, Model reduction via balanced state space representations, IEEE transactions on automatic control AC-27 (2), 1982, pp. 382-387.

[59] Rissanen, J., Modeling by shortest data description, Automatica 14 (5), 1978, pp. 465-471.

[60] Rissanen, J., Stochastic complexity and modeling, The annals of

288 References

statistics 14 (3), 1986, pp. 1080-1100. [61] Rosenbrock, H.H., State-space and multivariable theory, Wiley, New York,

1970. [62] Shibata, R., Asymptotically efficient selection of the order of the

model for estimating parameters of a linear process, The annals of statistics 8 (1), 1980, pp. 147-164.

[63] Silverman, L.M., Realization of linear dynamical systems, IEEE transactions on automatic control AC-16 (6), 1971, pp. 554-567.

[64] Slepian, D. (ed.), Key papers in the development of information theory, IEEE press selected reprint series, IEEE, New York, 1974.

[65] Sorensen, H.W., Kalman filtering: theory and application, IEEE press selected reprint series, IEEE, New York, 1985.

[66] Stewart, G.W., Error and perturbation bounds for subspaces associated with certain eigenvalue problems, SIA~I review 15 (4), 1973, pp. 727-764.

[67] Stewart, G.W., Introduction to matrix computations, Academic Press, New York, 1973.

[68] Tether, A.J., Construction of minimal linear state-variable models from finite input-output data, IEEE transactions on automatic control AC-15 (4), 1970, pp. 427-436.

[69] Theil, H., Principles of econometrics, Wiley, New York, 1971. [70] Willems, J.C., Least squares stationary optimal control and the

algebraic Riccati equation, IEEE transactions on automatic control AC-16 (6), 1971, pp. 621-634.

[71] Willems, J.C., System theoretic models for the analysis of physical systems, Ricerche di automatica 10 (2), 1979, pp. 71-106.

[72] Willems, J.C., Input-output and state-space representations of finite-dimensional linear time-invariant systems, Linear algebra and its applications 50 (1), 1983, pp. 581-608.

[73] Willems, J.C., From time series to linear system, part I: Finite dimensional linear time invariant systems; part II: Exact modelling; part III: Approximate modelling, Automatica 22 (5), 1986, pp. 561-580; 22 (6), 1986, pp. 675-694; 23 (1), 1987, pp. 87-115.

[74] Willems, J.C., Models for dynamics, in Kirchgraber, U., and H.O. Walther (eds.), Dynamics reported, volume 2, Wiley and Teubner, 1989, pp. 171-269.

[75] Willems, J.C., and C. Heij~ /2-systems and their scattering representation, in Bart, H., I. Gohberg and M.A. Kaashoek (eds.), Operator theory and systems, Operator theory: advances and applications, volume 197 Birkh~.user, Basel, 1986, pp. 443-448.

[76] WiUems, J.C., and C. Ileij, Scattering theory and approximation of linear systems, in Byrnes, C.I., and A. Lindquist (eds.) Modelling, identification and robust control, North-Holland, Amsterdam, 1986, pp. 397-411.

[77] Wolovich, W.A., Linear multivariable systems, Springer, New York, 1974.

S y m b o l index

This index contains symbols which are used more than local ly in this

monograph. The other symbols are explained in the nota t ion paragraphs in the

text . The numbers indicate the pages where the symbols are defined.

s e t s

N 34

Z+ 58

Z 32

~+ 15

Co~C+~C_ 97

(Rq) z 32

l~ 94

p o l y n o m i a l s

Rg×q[s,s -1] 34

Rlt×q[s,s -1] 40

d(r) , d(R) 39

~(r), ~(R) 51

c(R) 76

e(R) 39

mode l s mode l o b j e c t i v e s

B 31 c 22

B t 4 0 C D 127

B(R) 35 c P 135

BT(R ) 52 ¢ 22

B ± 3 6 e D 128, 146

B f 40 eP 136, 150

B s 46 e D 128, 144

RB s 49 e P 135, 148

Bs(A,B,C,D) 46 u 23

B](A,B,C,D) 96 Cto z 23

c(B) 91 eto z 27

ct(B ) 91 UCtot 23, 152

e*(B) 91 Ueto~ 27, 155

re(B) 91 * 15,1 U V to l

n( B) 91 e( Ctol) 159

data

w 31

143

q 32

2" 142

D 22

mode l c l a s se s

N 22

~3 174

[] 33

[]T, BT 51

S T 76 N , []T 84

[]2 95

Be 96

• ( T ) 151

[]~T' s 64 []PT

290 Symbol index

p r o c e d u r e s

P 22

Pu 23

Pctol 24

Ctol 132, 153

PP 140, 153 c tol

P e tol 28

P e tot 28

I~e to z 132, 156

P~tol 140, 156

~D 133, 157 ¢ to/

~etol 142, 157

P~ 57

p K /~K 70, 73

pO 76

P*, P* 81, 84

PT 63

im(PT) 64

misce l lanea

G

0 "-1, 0"*

a(A) #(v) #(7) A~±A2

col

[q,td Ch. I I I

HT(W) L(w) R(w)

ENT, ent

Ch. IV + ++

Z- - , Z- , Z , Z

L+, L_

(A+,B+,C+,D+)

(A_,B_,C_,D_)

K+

O_+ R

V

X

gr(F)

Ch. V

V t

Lt, Vt

L

Ft ?q j-)

S(~,d) A

32

34, 98

97

57

174

43

36

5O

71

8O

8O

72

94

99

103

105

102

104

112

49

46

46

97

4O

41

42

44

44

151

151

172

Subject index

A

algorithm for approxhnation by balancing for L+ for L_

for 1~cta

for

for I~cta for I~to l for ~tol

anticausality approximation

model optimal

autoregressive system

B

balancing behaviour

C

canonical correlation canonical form

descriptive minimal predictive

causality compatibility complementary space completeness complexity

descriptive of dynamical systems map predictive

consistency controllability corroboration

principle strong, weak

D

degree deterministic

113 104 105

162

164

166

166

169

170

97

89 174 35

111 31

139 37 42 37 44 97 78 40 33

127 91 22

135 175 96

22 64

42 4

driving operator driving variable

E

empirical covariance matrix equation error equation structure

corresponding to a complexity

equivalent parametrizations exact ness external variable

G

gap genericity

A-genericity

I

identification impulse response

stable incomplete tlankel array index input

L

lag structure law lexicographic ordering linear

procedure system

M

minimax property minimum description length misfit

descriptive of a law of a model

map predictive

of a law of a model

misspecification modelling

approximate deterministic

100 46

131 128 39

159 36 66 46

109 63

172

1 58

196 71

101 48

39 35 93

68 32

130 29

144 146 22

147 148 172

126 4

292 Subject index

exact 56 static 127 stochastic 4 under complexity constraint 23 under misfit constraint 27

monotonicity 66 bilateral 66

most powerful unfalsified model 60

O

order 39 output 48

P

parametrization 37 perfect observabili ty 47 Pontryagin

isometry 101 space 101

predicted functional 44 prediction polynomial 44 predictive law 44 procedure 22

corresponding to a utility 23 for deterministic time series analysis 143 identification 1 partial realization 70

prudence 64

R

row proper bilaterally zero order bilaterally

saJnple ra te scaling scattering representation

backward forward

scattering theorem selection rule sensibility shift shift invariant

procedure system

signal to noise rat io simplicity principle simultaneous equation model singular value

decomposition specification state variable stochastic system

AR backward state space controllable deterministic dynamical finite time generating 12 state space

realization backward 49 T input /s ta te /output 48 12 96 minimal 46 forward 46

realization problem minimal 57 partial 59

relative mean prediction error 147 remarkabili ty 76 representation U

autoregressive 35 shortest lag 39 tightest equation 39

Riccati equation 102 robustness 187

time invariance time reverse operator time series analysis total least squares translation invariance truly t - th order laws truthfulness

undominated unfalsified model unimodular matrix utility

52 39 52

197 203

99 99 98 73

159 32

67 50

193 22 25

130 130

2 46

4

35 49 96 31 31 50

174 95 46

32 49

143 133 50 41 69

57 36 23

Lecture Notes in Control and Information Sciences Edited by M. Thoma and A, Wyner

Vol. 81: Stochastic Optimization Proceedings of the International Conference, Kiew, 1984 Edited by I. Arkin, A. Shiraev, R. Wets X, 754 pages, 1986.

Vol. 82: Analysis and Algorithms of Optimization Problems Edited by K. Malanowski, K. Mizukami VIII, 240 pages, 1986.

Vol. 83: Analysis and Optimization of Systems Proceedings of the Seventh International Conference of Analysis and Optimization of Systems Antiba, June 26-27, 1986 Edited by A. Bensoussan, J. L. Lions XVl, 901 pages, 1986.

Vol. 84: System Modelling and Optimization Proceedings of the 12th IFIP Conference Budapest, Hungary, September 2-6, 1985 Edited by A. Prekopa, J. Szelezs&n, B. Strazicky Xll, 1046 pages, 1986.

Vol. 85: Stochastic Processes in Underwater Acoustics Edited by Charles R. Baker V, 205 pages, 1986.

Vol. 86: Time Series and Linear Systems Edited by Sergio Bittanti XVII, 243 pages, 1986.

Vol. 87: Recent Advances in System Modelling and Optimization Proceedings of the IFIP-WG 7/! Working Conference Santiago, Chile, August 27-31, 1984 Edited by L. Contesse, R. Correa, A. Weintraub IV, 199 pages, 1987.

Vol. 88: Bruce A. Francis A Course in Hoo Control Theory Xl, 156 pages, 1987.

Vol. 88: Bruce A. Francis A Course in Hoo Control Theory X, !50 pages, 1987.

Corrected - 1st printing 1987

Vol. 89: G. K. H. Pang/A. G. J. McFarlane An Expert System Approach to Computer-Aided Design of Multivariable Systems XII, 223 pages, 1987.

Vol. 90: Singular Perturbations and Asymptotic Analysis in Control Systems Edited by P. Kokotovic, A. Bensoussan, G. Blankenship Vl, 419 pages, 1987.

Vol. 91 Stochastic Modelling and Filtering Proceedings of the IFIP-WG 7/1 Working Conference Rome, Italy, Decembre 10-14, 1984 Edited by A. Germani IV, 209 pages, 1987.

Vol. 92: L.T. Gruji6, A. A. Martynyuk, M. Ribbens-Pavella Large-Scale Systems Stability Under Structural and Singular Perturbations XV, 366 pages, 1987.

Vol. 93: K. Malanowski Stability of Solutions to Convex Problems of Optimization IX, 137 pages, 1987.

Vol. 94: H. Krishna Computational Complexity of Bilinear Forms Algebraic Coding Theory and Applications to Digital Communication Systems XVlII, 166 pages, 1987.

Vol. 95: Optimal Control Proceedings of the Conference on Optimal Control and Variational Calculus Oberwolfach, West-Germany, June 15-21, 1986 Edited by R. Bulirsch, A. Miele, J. Stoer and K. H. Well XlI, 321 pages, 1987.

Vol. 96: H. J. Engelbert/W. Schmidt Stochastic Differential Systems Proceedings of the IFIP-WG 7/1 Working Conference Eisenach, GDR, April 6-13, 1986 XII, 381 pages, 1987.

Lecture Notes in Control and Information Sciences Edited by M.Thoma and A.Wyner

Vol. 97: I. Lasiecka/R. Triggiani (Eds.) Control Problems for Systems Described by Partial Differential Equations and Applications Proceedings of the IFIP-WG 7.2 Working Conference Gainesville, Florida, February 3-6, 1986 VIII, 400 pages, 1987.

Vol. 98: A. Aloneftis Stochastic Adaptive Control Results and Simulation Xll, 120 pages, 1987.

Vol. 99: S. P. Bhattacharyya Robust Stabilization Against Structured Perturbations IX, 172 pages, 1987.

Vol. 100: J. P. Zolesio (Editor) Boundary Control and Boundary Variations Proceedings of the IFIP WG 7.2 Conference Nice, France, June 10-13, lg87 IV, 398 pages, 1988.

Vot. 101: E E.Crouch, A. J. van der Schaft Variational and Hamiltonian Control Systems IV, 121 pages, 1987.

Vol. 102: E Kappel, K. Kunisch, W. Schappacher (Eds.) Distributed Parameter Systems Proceedings of the 3rd International Conference Vorau, Styria, July 6-12, 1986 VII, 343 pages, 1987.

Vol. 103: E Varaiya, A. B. Kurzhanski (Eds.) Discrete Event Systems: Models and Applications IIASA Conference Sopron, Hungary, August 3-7, 1987 IX, 282 pages, lg88.

Vol. 104: J. S. Freudenberg/D. P. Looze Frequency Domain Properties of Scalar and Multivariable Feedback Systems VIII, 281 pages, 1988.

Vol. 105: Ch. I. Byrnes/A. Kurzhanski (Eds.) Modelling and Adaptive Control Proceedings of the IIASA Conference Sopron, Hungary, July 1986 V, 379 pages, lg88.

Vol. 106: R. R. Mohler (Editor) Nonlinear Time Series and Signal Processing V, 143 pages. 1988.

Vol. 107: Y. T. Tsay, L.-S. Shieh, St. Barnett Structural Analysis and Design of Multivariable Systems An Algebraic Approach VIII, 208 pages, 1988.

Vol. 108: K. J. Reinschke Multivariable Control A Graph-theoretic Approach 274 pages, 1988.

Vol. 109: M. Vukobratovic/R. Stojic Modern Aircraft Flight Control Vl, 288 pages, 1988.

Vol. 110: In preparation

Vol. 111: A. Bensoussan, J. L. Lions (Eds.) Analysis and Optimization of Systems XlV, 1175 pages, 1988.

Vol. 112: Vojislav Kecman State-Space Models of Lumped and Distributed Systems IX, 280 pages, 1988

Vol. 113: M. Iri, K. Yajima (Eds.) System Modelling and Optimization Proceedings of the 13th IFIP Conference Tokyo, Japan, Aug. 31 - Sept. 4, 1987 IX, 787 pages, 1988.

Vol. 114: A BermL]dez (Editor) Control of Partial Differential Equations Proceedings of the IFIP WG 7.2 Working Conference Santiago de Compostela, Spain, July 6-9, 1987 IX, 318 pages, 1989

Vol. 115: H.J. Zwart Geometric Theory for Infinite Dimensional Systems VIII, 156 pages, 1989.

Vol. 116: M.D. Mesarovic, Y~ Takahara Abstract Systems Theory VIII, 439 pages, 1989