Time Domain Left Invertibility: Application to Failure Detection

Copyright © lFAC System, Structure and Control Oaxaca, Mexico, USA, 8-10 December 2004

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TIME DOMAIN LEFT INVERTIBILITY: APPLICATION TO FAILURE DETECTION Moises Bonilla Estrada '.1 Maricela Figueroa Garcia ".2 Michel Malabre .... 1

• rINVESTAV-lJ'N, Control Autom.atir.o. AP 14-740. Mexico 07UUU, MRX fro. mbonillll{fJJcin'Ul'l>tll'V. TfIX . •• PhD student of J. C. Mal'tinez and M. Bonilla at Cf NVESTA V-I P N . mfigueroa tkct1'l. cinvestav. mx . ••• lRCCyN. H.P. 92101. 44.'121 NANTb'S, Cerle:r OS, FRA NCR. Michel. Mul(~bl·e@·il·ccyn. ec-nuntes.f)·.

the l'iyht inve!'se and the left inuel'se system.s. A right inverse system is used to control (e.g. to reject dist.mbances) , whilf' a left im'erse s1/.~tem is nsed to ohserve (e.g . t.o reconst.ruct particnlar signals present ill the system).

Abstract: We cow;iuer here the problelll of recovering from the output alone. and at any time t :::: O. t.he (unknown) input, with init.ial condit.ions not. necessarily ef[nal t.o 7,ero. In this casf' left inve1'tibility of tmn4er functions is [lot sufficieut. Copyright ©2004 fFAC.

III this paper, wc study of the left inVC1'SC .systems with applications to the failure detection problem (see Figueroa et al. (2004)). We want to solve:

NOTATION

Script capitals V,)tV, . . . , denot.e linear spaces with elements v , w, . .. ; whe[l V C W. ~ or W /V stands for the quotient space W modulo V; the external direct sum of some given spaces XI,· .. , Xr is wriltf'n as Xl.;B · · ·Xr . Given a map X :V .... W. Im X == XV dellotl.'S its image, Ker X dellotes its kemeL X ~ I T the iuverse image of T by the linear map (possibly not invertible) X. XT denotes the transpose of matrix X. ei stands for the vect.or havillg a 1 in its i-tit coordinate and 0 in the other ones. {VI, . .. t'k} sta[lds for the subspace generated by the vectors VI, .. . Vk. :i; and i denote t he first. and Sl'cond t.ime derivatives of x, x(i) is used for higher orders derivatives. p and s stand for d/dt and the Laplace variable. respectively.

Problem I. Let us cOlIsider the folluwillg state space description, 1:(A, r, C): W -> Y: i: = Ax

(1)

2. LEFT INVERSE Remark 2. :

Rl The basic definition for left invertibility is: Given a function f: Dom .... CoDom. r >-> f(r), find (if possible) a fnnrtioll g : CoVom .... Dom, '1' ....... y(-t», such that the composite function go f ; Dom .... Dom, r ....... g(f(r)). is the iJelltity function I. Namely: g(f(r)) = r for all r EDam.

Two powerful tools usually used in System Theory for synt.hl'Sis and analysis are (whf'n t.hey exist)

1

Y = Cx

ClIuer which cOllditiulls does there exist, alld , then. how to design a left inverse system for (1), independent. of bot.h: the initial conditions of x and th(' nature of inpnt 11' .

1. INTRODUCTION

2

+ rlL'

whf're w is t.he input., y is the ont.pnt and x is thf' stat.e. The linear maps arc defined a.~ i i:X .... X, r: W .... X, and C: X -+ Y: where X, W, and Y are the state, input, and output spaces.

Lab. Franco-Mexicain d'Automatique Appliqut'e. Sponsored by CONACyT~Mexico

573

R2 For a linear function f. the existence of a

with III '" o. The external behavior of this system is gil'P.TI by the ordinary dijJerent.inl equation:

lp.ft im'p.r8p. fundion. g. is ff(lIivalent to the fact that f has to be munk. Namely: Ker f(-) := {r E Dam I fer) = O} = {O} .

p2y = (p

and its Transfer Function Matrix,

R3 From the funuamental Theorem of Culculus: The left inverse function . g. of the linear function

f : r(t)

~: r(T)dT

1-+

it~: r{T)dT

is: g: vel)

1-+

Ud~l).

+ ')'o)w

l{1,(s) = (s

since

(7) l~(s),

The information inl'Olllf!d in ThP.Ofem

= r(l) , for all integrable variable r .

is: (~)

+ lO)/s2

4 is:

Im r = {el +IOe2} andKer C = {e2} = Vf-, then Vii, + 11Il r = {ell + {Cl + ')'OC2} = {el, C2} ' This 'i mpl'it;l:!: A - 1(Vi-+Im r) = A - I {el.ed = {el.e2} , and thus Vf = {e2} n {el,e2} = {e2} = Vf; then :

In the study of Linear Time Invariant Systems, then' are basically t.wo domain.~. Namely. the fn:qut:.nC::t! dumain awl the time dUT/win. III the fullowillg two suO-sectiolls we recall SOllle useful results about left invertibility in each domain.

Vr = {e2} Ker

r

=

and

{O}

lIu

and

r

+ ~(lIe2} (Y) r n Vr = {O} (10)

= {el

Im

Therefore . system (fi) is TF~l left invprtihlp. Indeed, fmTl! (8) itl:! TFr.I left i11verse, T~i' (S), is: T;(s) = 52/(5 + ~(o) (11)

2.1 Transfer Function Matrix Left Invertibility 'vVhell uealillg with a tmn:;fer funci'ion TIIat7'ix (TFM). the following definition for left invertibilit.y is st.andard (c./- with Remark 2:R2) :

which is the Laplace transform of the system:

(p

Definition 3. Let \IS consider a transfer function matrix. '1'(5) = C(sI - A)-Ir. with p rows and m columns. 1'(s) is TFIII left inl'p.rtible if and uuly if its ut colulllns are inuepelldent as rational functions of s, namely if and only if rank T(s) = m. viz. if and only if Ker 1'(5) = {O}.

Rcmark

(j.

+ ')'o)liJ =

))2y

(12)

:

RI From (I') ancl (11) : T:1,(5)' T;:(5) = I, b1\t

R2 From (7) and (12): (p +

/0)(111 - w) = O. \Vhich implies: ';; = w + ke-~ot, moreuver

lII{O)P-~ot. with w{O) non ?;ero, then frulll (7) we get : y = 0, alld thu8 W = o.

R3 If: 111 = This TFM lefl im1erlibility was geomet.rically characteri
V'f = X,

Vj.+!

= Kel'

C n A- 1 (Vj.

A nd thus, the TFl\l invprt.ibili ty depends on init.ial condit-ioll.s as well a.s on t/tI; natur'e uf w .

Indecd, fmfIL Remark (j:R2, WI:. nel:.d t/wt the initial condition k be in a neighborhood of zem with a very small radius. And, which is more im.portant, the parameter 10 m.1J..~t be non nel]ati1Jp.. Furthl:.l'1fL01·1:., fmTII Rernar'k (j' :R3, tltt:. input w c(mnot belong to Kel' (p + 10) '

+ 11111') (2)

Thl'.Drem 4. (13asile ami Marro 1992) 1'{s) is TF!Il left iTl'Uatible if and uuly if:

r

Ker

= {O}

and

Im

r n Vr

= {O}

(3) 2.2 Time Domain Left Invertibility

Let us note that the geometric characterization (~) is also e(l'livalent to the following one: Ker l' = {D}

and

nr =

{O}

As illustrated in Remark 2:R3, a state space description usually has non proper left inverses. 'vVe t hen have to work in t he more general framework uf implicit systems (SL'C for example Lewb (l9!)2)). In the time domain (TD). we take the following definition for left invertibility (c.f. Remark 2:RI):

(1)

nr

where is the supremal (A. r) cOlltrullability subspace contained in Ker C (see Wonham (1985) for more details), which is the limit. of

n~ = {D}

n~+1 = Vr n (An~

+ Im

Definition 7. The system (1), E(A, r, C), is called TD left in1Jp.rtible if and only if: (i) therp pxists a system Ei : y ...... W such that it is solvable :l ill Im E and (ii) Ei 0 E = I. Ei is railed a TD left imlP.r.~e of ~.

1') CS)

In ord<>r t.o show the spedfidt,Y of TFM left illversel:!, let \IS cunsider the folluwiug example: Example S. (Start)

Let

ItS

r.onsider th.e system:

Y=[lO]X

3 Solvahle means that for each input in y, there exists at least one solution in X .

(6)

574

:'\ote that, for the solvable implidt system (1) , = Ax + ru and y = Cx, there exists a linear transformat.ion IJI(·) : W -+ X. surh t.hat : l~(w) = AIJI(w) + rw ami y = (,'1JI(w). for all adl1Jissible inputs w . 130nilla and l-.Ialabre (l!)!)U) stated the following results (c.f. Remark 2:R2) :

which are always zero) . In the particular ca.-se of regular systems, F and C square. and det[.>.F Cl i' O. there only exist. finite and infinite element ary rli visors.

Ix

Wong (H)74) and Uf'rnhard (1982) havf' g(,()IIIPt.rically characterized the fiuite e1el1lentary divisors through the maximal (F. C) invariant subspace, VA·.IJ = sup {VeX ICV cFV}. which is thf' limit of thl' algorithm : Vk.1J = X , and V~~ll = C- I FV~O" If .>. is a finite-zero of the pcndl [>.FCl, there then exists an exponential mode characterized by a vector l' E V X.IJ sllrh t.hat Cv = '>'Fv .

Lemma 8. (Bonilla and Malabre 1990) The implicit description (1) is 'l'D left invertible if and only if Kf'r CIJI (.) = {O} . Lemma.9. (Honilla and Malabre 1990) If t.he implicit descriptioll (1) is TD left muer'lible thell OIlC left illverse systelll, ~i : y -+ W, is: -I

lE i .; where lE,

=

= Ai'; + Eiy

[~~

l

Ai

=

ill

= C i .;

[~~ ~ ],

E,

Armentano (1986) has geometrically characterized t.he global Kronffker demmposition. In t.he ('a.<;(' of a rl'gular pendl thl' finitp dl'mf'lltary divisors [>'1 - Jd (J i being the corresponding Jordan matrices) are located in the restriction of [.>.F Ul to V".IJ in t.he ciomain and t.o FVY.t1 in t.he ('odomain.

(13) [

~I]

and C, = [0 I]. In this case the linear maps are defined as: lE, : Xe -+ X p , Ai : X. -+ ,rc' E, : Y -+ X and C i : X, -+ W: where X, = X EB W and.a:., = Y lOB X .

Relat.ed with system (l:l) is t.he sl1premal (lEi. A;) invariant su\)space. VX•.II = sup{V e XclAi V elE, V}. which is the limit of the non increasing algorithm : ,-~I -,<+I A-llE V-,< V VX • . II = Xe Xr. ,O = i i ..l'f': ,n

Let. us not.e t.hat" if r is monif' and if (1) is ohsf'rvahle thf'n t.he proposed TD IPjt inverSE (13) is minimal under extemal equivalence (see Kuijper (1992)). Indeed, (i) the matrix [lE; Ri] is epic, (ii) the matrix

[""EiC~ Ai]

[~:]

Let llS not.e that. a necessary mndition for (n) to be a T D ll'.ft invcm ;! of (1) is that V,i.• ,11 = {U} . Indeed, if this is not the case, problems occur with the initial conditions for the exponential mocies (int.egrat.ors) , rharact.erized by V,~,.II (r.I Armentano (1986)) and in general ~i (~ (-)) will not be the identity operator.

is monic and (iii)

has full column rank for all complex

nUlllber .>. if and only if this holds for matrix [ .xI

~.4],

namely: if and only if N = {O} ; where

N is the unobservable subspace of the (A, C) pair, i.e.: N = n~';olA-iKer C .

3. GEOl\IETRIC CHARACTERIZATION OF TD LEFT IN\lERTIBILITY

2.2.1. Exponential modes We know from Gantmacher (1977) that for allY pelldl [.>.F - Cl, with F : X -+X and C : X -+X, there exist bases of X and X such that its associated block diagonal matrix (known as the Kronecker decomposition) is LVl1Jposcd of four types of hlocks, nameiy: (i) fillite elementary divisors, (ii) intillite elementary divisors. (iii) column minimal indices. and (iv) row minimal indices .

In this Sfftion we propose a geomet.ric chararterization of TD Left lnl'ertibility. Theorem 10. The state description (1) is TD left inllertible if and only if

Ker

The finite and infinite elementary divisors correspond to thf' proper (differential equations) and the nOli-proper (dcrivators) parts, rL'Spectiveiy, of the system . The Lvlumn alld row miuimal indices correspond respectively to the existence of degrees of freedom (more unknowns than equations) and to the null part. of the system (state contributions

r

= {O}

and

Im

r

n (Vr

+ AW)

= {O} (14)

In order to prove TheorcJIl 10. we need the following two Lellllna.-s:.') Lemma 11. (Figueroa et al. 2004) If Im r n + AVn = Ker r = {O} then y = 0 implif'S w = 0, for all t 2: O.

(Vr

Lemm.a 12. (Figueroa et al. 2004) If Ker r = {O} thm : r- I (Vr + AV = {O} if V.\-, 11 = {O}.

r)

4 This left inller.• e is (in general) non minimal : however it can be easily minimized by matricial algorithms (see for

example Honilla and Malahr .. (1997)) . A pro("ctlllf(' ("an be

found in (Bonilla and ~1a1ahre 1994) . In that paper also is considered right invertibility.

5

L

575

See Lemmas 9 and 10 of l"igueroa et at. (2004) with = r , U = 0 and m = w.

Proof of Theorem 10

To prove ThPOrem 14 we neen t.hree Lemmas: I;

Let us first prove the sufficiency: For this let us suppose that Ker r={O} and Im r n (Vr + AV,-,) = {u} . Thcn by LClIIllla It, y = U ilflplies w = 0 \;/ t :::: 0, and thus, frOIfl Lelfllfla 8 the system (1) is TD left invertible. Let us now prove the necessity: If Ker r i {O} t.here t.hen exist.s a w i 0 snch that rw = 0 which implies that there exists a non zero w which is non observable at the output, and so, Ker C\lI(w) i {D} . Therefore r has to be monic .

Lemma 15. (Fig1\eroa et al . 2004) (Vr + AV n IIll r = {U} and Ker r = {O} if and ouly if Vr = Nand r- 1 (N + AKer C) = {O} .

r)

Lemma 16. (Figueroa et al. 2004)

r

(1) V~ = IlV (2) r- 1 (N + .l1Ker C) =

r- I .1Ker C .

Lt;7Itmu 17. (Figueroa et al. 200·1) Vr = Nand C) = {O} if and only if = {O} and Ker = {O} .

W

r - I (N + AKer

In order to prove the necessity of the second geometric condition. let 1\S as::;1\me t.hat syst.em (1) is TD left in'Ut;riible. thcn system (13) is one TD left i.,Wt;7·St;. If system (13) is a TD left inverse then it has no exponential modes, and t hlls V,\- 11 = {O}. Therefore by Lemma 12, WP get. t.he seconn geomet.ric connition. 0

r

Proof of Theorem 14 From Lemma 15, (Vr + AV n Im r = {O} ann Ker r = {O} hold if ann only if Vt = Nand r- I (N + AK('r C) = {a}, and frolll Lelllllla 17 if and only if V~ = {U} alld Ker f = {O}. 0

r)

Let us come back to Example 5

Example 13. (Cont'i7lued)

4.2 Zp-ro StT1/elure

F1'OTIL (9) we get:

In the 70'::;, pioneered by the work of Ilosenbrock

Im

r n (W + AVf) = i

{e1

+ fOld n ({ e2} + {I'd)

{O}

(l.'i)

(197U), there was a great illterest about the zero structure of lineal' lIlultivariable systellls (see fur example MacFarlane and Karcanias (1976) and Francis ann Wonham (H)75»). Lat.er, Aling ann Schumacher (1984) pruposed a complete geollletric characterization of the zeros structures.

thPen , thPe .I1comPetric condition (14) i.~ not 8atisfied. And titw;, system (6) i.s not TD left illvertible.

III this paper we shall be concerned with two particular subsets of the invariant zeros, namely the tran.smission zeros and t.ht' inp7Lt dero7Lplin..I1 invariant zer08 (see Aling ann Schllmacher (l!)~1) for COIll plemellts).

4. STRUCTURAL CHARACTERIZATION OF

I'D LEFT INVERTIBILITY In this Section, we give a structural characterizatioll of TD Lt;ft In'Ut;rtibility which is equivalent to those of Theorem 10. For this, we first extract a maximal observable part of the system, and we next rpcall some rPSlIlts abo1\t t.he zero structme.

From (Aling and Schumachcr 1!J81), thc total number of transrni.s.sio71 ze7'OS alld input decoupling invariant zeros, here called observable invariant zeros (for shortness), is:

. :;eros # obs . tnv.

{ I Ma:r.imal Obserl1able QuotiP.nt SY8tPem

Let 11 : X -> X/ N be the canonical projection , there then exist unique maps (A, r , C) such that:

1I11=AlI;

lII'=r;

C=ClI

V~+l = Ker CnA-I

(18)

{3 StT1/elural Characterization

(16)

Corollary lR. The st.ate opscript.ion (1) is I'D lp-ft invertiblPe if and only if it IS TFM left invertible and it has neither transmission zero.s, nor input decoupling invariant zero.s.

Let V~ he the sllpremal (A, f') invariant slIhspace contained in Ker C, namely V~ = sup{Vr C Ker G' I i:tVr C Vr + Im r}; which i::; the limit of the non increasing algorithm (for J1. :::: U) :

~ = X/N,

Vi = d'lm :::;-:---"---;-;: n~+N

(?r + Im r)

PROOF.

( 17)

Let us first suppose that (1) is TD left iTlvt;1'iible.

Theorem 1{ Given the system defined by the maps (A . CC), let the q1lotient system be nefined by the maps (A, C) . Then (VI~ + AVnnIm r = {O} and Ker r = {O} if and ollly if = {O} and Ker = {O}.

Then Theorem 10 implies that Ker r = {O} and Im r n Vr = {O} . This last geometric condition

r,

r

W

6

L

576

See Lemmas 12, 13 and 14 of Figlleroa et a/. (2004) with = r, L = t and M = W.

'Rr

hi equivalent to Itn r n = {O} , and from algorithm (5) we get Rr = to}. Since Ker r = to} and = {O}. WE' concludE' from ThrorE'm 4 t.hat (1) is TFM left inllertible (recall the equivalence (4)) . On the other hand, from Theorem 14 we get: V;' = Ker n = N , which impliE'S: dim (V;' / (R + N)) = dim (vt/N) = to} . And then, from (18), thf'rf' arc neither tmnsmtssto71 ze1'OS nur i1lput decoupli1lg invariant zeros, i. e. no observable invariant zeros.

In section 2 we have pointed out that left invertibility is efJllivalenl to monicity (c./ Remark 2 R2 ,mu Lemma 8) , Then, (22) will be left iuvertible

'Rr

if and only if [ ~)

Let us suppose that (1) is TFM left invertible and it does not have any observable invariant zero.

r

liS

To gel. a TO left inverse, let ;i:

come back to Example 1:1

~ = [~ ~ ] € +

[n

10

;

lU

w ; y

=

[0 1] (

= [1 0] €

and since

Let us consider the following system

(2U)

[,;] ;

[~]

(23)

we get

H

"

(2!))

We have proposed in this paper necessary and sufficient conditions for left inversion when working in the time domain, We have pointed Ollt that in the time domain the initial conditions play an important role for left i1IVe1'tibility. As shuwn un uur exalJlple, which is TFM left invertible, there indeed exists an inpllt which is not observable at t.he ontPllt. and thns which cannot he reconstrnctf'd, nf'ither by illversiulI tech11 iques, nur by allY observatiun techniques,

7 .

y = Cx

u == u

6. CONCLUDING REMARKS

(21)

where u is the input , y h; the uutput, x is the stat.E' and m is a given fail me. The linear maps arc definf'd as A:X -+ X, B : U -+ X , L : M -+ X ,md C : X -+ Y : where X , U , M and Y are the state, input, failure and output spaces. To reconstruct m (Saberi et al. 2000) , using inversion tcchniqllcs , let lIS rewrite (21) as: i: = Ax+ [ L B)

rewrite (21) as :

= Ax + [L B) [ ~] ; y = Cx

lJ

5. APPLICATION TO FAILURE DETECTION

+ Bll + Lm

IlS

' _rCO] [-IU][Y] [uu] l()~:t:ll/+

(19)

Indeed, the external behaviours of (19) and (20) are, respectively, described by the ordinary differentia/ equations: w = (p + I'lI)W and p2y = W, System (20) is TD left invertihle, (it has no ubservable zero), one TD left inven;e being: ill = p2y, And in this way, we can obtain , by inversion techniques, I,he fill,ered input 1/', but not 1V ,

i: = Ax

0,

Then (in case of TU left in1lertibilily) we get from Lemma!) the failure rewlll;trudur cI (13)) ,

B.WT1l11Ie 19. (End) In 1Iiew of Corollary 18, Ule 7mlize that system (6) is not TD left inveltihle due to the presence of the observable zero (s + /0) (see (7)). One possible solution to overcome th.is can be, for e:rnmple, 1,0 decompose (6) as th.e cascade of the following L-ine(l7' systems:

' (+ [-1] 10 (= -1'0 [00]

['~I] =

Corollary 20, The failure m of system (21) can he reconstructed hy left invcrsion tcchniques if alld ullly if C(sI - A) - l D is TFM left invel'tible and it has neither transmission zeros, nor input decoupling invariant zeros,

Then from (4) and (18), we have Ker r = to}, Rr = to}, and V;' = N . Now, since N is AinvarillIlt. , wc get : Im r n (V;' + AVf.) = Im r n V Since Ri· = to} we have: Im r n V;' = tU}. Therefore, Ker r = to} and Im rn (V + AV;') = to}; and thus, Theorem 10 implies that (1) is TD left invertible. 0 LE't

0 implies that

That is tu say, i: = Ax + Dm alld 0 = rx, illlply that m = 0, This is equivalent to ask TD left in1ler/,ibilily of (1), with r = Land w = m. We then have the following Corollary (c.f Theorem 10. Theorem 14 and Curollary 18) ,

r

r·

=

[ ~,] x+ [ ~ ~ ][:' ]

In Hou and Patton (1998) is introduced a notion which is very close to TD left invertibility. namely, t he one called inpul ob.~enJabilil, y Ij • In that paper the input observability is characterized ill a matricial way by their Theorem 1, where is stated as a necessary and sufficient condition for input

(22) DirTerent. from the (weaker) lJotion of input obsenJability used in Massoumnia et al (1989) which corresponds to: r monic and lm r n N = to},

8 7

See also (Figueroa et ai, 2(04)

577

ob~t:n}(lbility

D

the following equality (in our paper

Basile. G. and G . Marro (1992) . Cuntmlled and Conditioned Invariants in Linear System Thp.ory. Prentke Hall. I3crnhard, P. (HIR2) On singular implicit d~' nam­ ical systel11s. 81A M Journal on (;ontml (md Optimiz.ation. 20(5) , 612- 633. Bonilla, r-.I. and M. Malabre (1990) . One side invertihility for implicit. dl'Scriptions. In : 29th C07Lfcr'c7Lce on Dec i~ion and (;o7Ltml. pp. 3601- 3602 . Bonilla, l\I. and M . l\Ialabre (1994) . Geompt.rir Charact.t'ri7.at.ioll of Lewis's St.rIlctnre Algoritlllll . Cir'cuits , 8y~ tclfLli und , >'ig7Lul Pr'occlis ing, special i.~sue on "Implicit and Robust Systems" 13(2-3) , 255- 272 . I3onilla, M. ann M. Malahm (1997) . St.mctnral Matrix :'v1illil11izatioll Algoritlllll for hnplicit Descriptions. Automatica 33(4) , 705- 7lO. Francis, B.A. and W .M. Wonham (1975). The role of transmission 7.eros in linear mn\t,i variable regulaton;. Intc7"1wtiunal JUU17lUl of Contrul, 22, 657- 68l. Figueroa, M ., M. Bouilla, :\1 . Malabre and .l .C . :\'lartine7. (2004). On failllrp deteetion by inversion techniques. To be presented at the 43r'd Confer'wct: 011 Dt:cision anti Control. Gantmacher, F .R . (1977) . The Theory of 1latriCl'S. Vol. Il , New York: Chelsea. IIOll , M. and R.J. Patton (l9!)8) . Input Observability and Input Reconstruction, A utumutiea, 34(6), 789- 794. Knijpcr. M. (1992) . Dl'Script.or reprl'Sent.ations without direct fcedthruugh tenn . Auto1TLatica. 28,633-637. Lewis, F .L. (1992). A tutorial on the geometric analysis of linear time-invariant implicit systems. AlLtomatiea 28(1) , 119 -1:17. MacFarlane , A.G.J . and N. Karcanias (1976) . Poles and zeros of linear multi variable systems : a survey of the algebraic, geomet.ric and complex-variahlt' theory. International Journal of (;01ltr'ol, 24 , 3374. :'.-lassoumnia r-.I.A ., G .C . Verghese. and A.S . Wilsky (1989). Failnrp Det.ection and Ident.ification. IEEE Tmn.m cfions on. /lu/.omatic Contml, 34(3), 316- 321. Rosenbrock, H.H . (1970). State-Space and Multivariable Theory. London: Nelson. Saht'ri A., St.oorvogd A.A., Sannuti P. and Nil'lIIallU H.H. Fundal11ental problems in fault detection and identification . International Journal of Robust and Non/inear Control, Vol. lO, pp. 1209-1236. Wong, K.T . (1974) . The eigenvalue problem )"Tx+Sx.,1974. Journal of Differential Equations, 1, 270-281. Wonham, \V .M. (198.')) . Line.ar MlItlillariable Contr'ol: A Geometric Appmach. 3rd ed .. Springer-VerJag. New York.

= 0 and r = B) :

where u()..}.-f - N) denotes the set of the finite pigenvalnes of t.hp ppncil )"M - N (t.h~ arp the finite elementary divison; recalled in Section 2.2.1) . But this cOlldition is ollly necessary alld 1I0t sufficient. Indeed, let us consider the example: X·

=

[~~ ~ ~l + [~ ~l 0001 0000

x

01 00

y=

tL

I () 11 0] [ 0010 X

(27) The first input, U\, is associated to the non observable snbspace {e2}, and t.he second inpnt. It'l' is linked to y'l by the ordinary differcntial equation ih = U2. It is clear that this system can not be neither TD left invertible nor input ohs en 'a hie. Compllting the sllbspacps involved in Theorem 10: Ker

r = {O} &

Im

rn(W+AVn = {e2' e:d

i- {O}

which implies t.he non TD left i1l.11ertibilily. Let ns consider t.he pencils liSI'd in Theorpm 1 of IIou and Patton (1!J!J8) , namely .HI ()..)

- <\ = [ AI c'

-D] D

all cl

TLunnal-mnh:!; of these rank(Ah ()..)) = 5 and Then : A= rankM\(A): { Ai-

1\{< () A

= [AI C- '<\] ·

Tlle

two pencils arc: normalnormal-rank(M2 (A)) = 4.

0 => rankM\ = 4 < 5 0 => rankM\ = 5 = 5

rankM ()..) : { ).. = 0 => rankAh = 3 < 4 2 ).. to=> rankM\ = 4 = 4 Which illlplies that : u(M\ ) = {O} = u(;\/~), alld then Theorem 1 of (Hou and Patton 1998) says that system (27) is input observable. This contradiction arrivps sincp condit.ion (26) is only npcessary and 1I0t sufficiellt. Hou and Pattoll (1998) have to add the condition of TFM invertibility, namely to add the condition R8 = {O} (see (4) and om Corollary 18). Indeed. in t.his academic example , (27) has a rank equal to 1 and not 2, i. c. the number of inputs; and it is precisely the input belonging to = {O} which makes problems.

Ra

REFERENCES Alillg, H. and J .M. Schulllacher (1984) . A nine-fold canonical decomposition for linear systems. Internal.ional Journal of Conl.rol, 39(4) , 779- 805. Armentano, V .A. (1986) . The pencil (sE - A) and controllability-obsprvability for generalized linear systems: a geometric approach. SIAM Joumal ur£ (;ontml and Optimizatiun, 24(4),616-638 .

578