MAXIMAL INVARIANT SUBSPACES AND OBSERVABILITY OF MULTIDIMENSIONAL SYSTEMS. PART 2: THE ALGORITHM

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1 U.P.B. Sc. Bull., Seres A, Vol. 80, Iss. 1, 2018 ISSN MAXIMAL INVARIANT SUBSPACES AND OBSERVABILITY OF MULTIDIMENSIONAL SYSTEMS. PART 2: THE ALGORITHM Valeru Prepelţă 1, Tberu Vaslache 2 The paper s connected wth the Geometrc Approach, a trend whch enrched the feld of System Theory wth new notons and technques. An algorthm s proposed, whch determnes the maxmal nvarant subspace wth respect to a fnte number of commutng matrces and whch s ncluded n a gven subspace. The complete proof of the algorthm s provded. Ths algorthm can be used n the study of the observablty of multdmensonal (nd) lnear systems and to determne the subspace of the unobservable states. A Matlab program s proposed, whch mplements the algorthm and computes an orthonormal bass of the maxmal nvarant subspace. Keywords: Observablty, maxmal nvarant subspaces, multdmensonal systems, dscretetme systems. MSC2010: prmary 93B07, 93C35; secondary 93C05, 93B25, 93C Introducton The Geometrc Approach s a trend n System and Control Theory whch has provded smpler and elegant solutons for many mportant problems, such as controller synthess, decouplng, pole-assgnment, controllablty, observablty, mnmalty, dualty, etc. The hstory of the Geometrc Approach started wth the papers of Basle and Marro (see [3]) and was developed by Wonham and Morse [10], Slverman, Hautus, Wllems et al. The cornerstone of ths approach s the concept of nvarance of a subspace wth respect to one lnear transformaton. In the past four decades a lot of publshed paper and books have been desgned to the theory of multdmensonal (nd) systems, whch has become a dstnct and mportant branch of the systems theory. The reasons for the ncreasng nterest n ths doman are on one sde the mportant applcaton felds (sgnal processng, mage processng, computer tomography, gravty and magnetc feld mappng, sesmology etc.) and on the other sde the rchness and sgnfcance of the theoretcal approaches, some of them beng dstnct from those concernng the theory of 1D systems. 1 Department of Mathematcs and Informatcs, Faculty of Appled Scences, Unversty Poltehnca of Bucharest, Romana, E-mal: valeru.prepelta@upb.ro 2 Department of Mathematcs and Informatcs, Faculty of Appled Scences, Unversty Poltehnca of Bucharest, Romana, E-mal: tberu.vaslache@gmal.com 13

2 14 Valeru Prepelţă, Tberu Vaslache Varous state space 2D dscrete-tme models have been proposed n lterature by Roesser [9], Fornasn-Marchesn [4], Attas [1] etc. Ths paper extends the Geometrc Approach technques to present an algorthm for determnng the maxmal subspace whch s nvarant wth respect to a fnte number of commutatve matrces and s ncluded n a gven subspace. It completes [8] by provdng the proof of ths algorthm and also by determnng recurrently the maxmal nvarant subspaces w.r.t. the frst one, two etc. matrces. When the matrces represent the drfts of a multdmensonal lnear system and the subspace s the kernel of the output matrx, ths algorthm can be adapted to determne the subspace of unobservable states, wth applcatons n the study of the propertes connected wth the concept of observablty [6]. The dual algorthm whch determnes the mnmal nvarant subspace that ncludes a gven subspace (e.g. the mage of the nput matrx) s descrbed n [7]. Secton 2 gves the descrpton of the maxmal nvarant subspace wth respect to r 2 commutng matrces whch s ncluded n a gven subspace. An algorthm whch calculates ths maxmal subspace s proposed and ts proof s developed. Ths algorthm s a generalzaton of the 1D method of G. Marro (see [5]). It was appled (wthout proof) n [8] to determne the subspace of unobservable states of a multdmensonal (rd) system. Secton 3 provdes a Matlab program that mplements Algorthm 2.1 presented n Secton 2 whch computes an orthonormal bass of the maxmal nvarant subspace. An example llustrates the advantages of the proposed method. In Secton 4 t s shown how Algorthm 2.1 can be appled to problems concernng the concept of observablty of a class of multdmensonal dscrete-tme systems. matrces. 2. Algorthm of maxmal nvarant subspaces Let K be a feld, C a proper subspace of K n and A 1,..., A r K n n commutng Defnton 2.1. A subspace V of K n s sad to be (A 1,..., A r )-nvarant f A v V, v V, {1, 2,..., r}. A subspace V of K n s sad to be (A 1,..., A r ; C)-nvarant f V s (A 1,..., A r )- nvarant and t s ncluded n C. (A 1,..., A r )-nvarant and ncluded n C, Ṽ V. V s called maxmal f, for any subspace Ṽ whch s Let us denote by maxi(a 1,..., A r ; C) the maxmal (A 1,..., A r )-nvarant subspace ncluded n C. V}, A k where ( r For a subspace V of K n, we consder the followng subspaces: A 1 = {v K n A k v V} and ( r =1 A 0 )V = V. If v A =1 A k )V = {v K n ( r =1 Ak = {v K n A v )v V}, k N, V, then A v A () V, {1, 2,..., r}, 1.

3 Maxmal Invarant Subspaces and Observablty of Multdmensonal Systems. Part 2: the Algorthm 15 Proposton 2.1. The maxmal (A 1,..., A r )-nvarant subspace ncluded n C s maxi(a 1,..., A r ; C) = ( A k )C. (1) k r=0 =1 Proof. Let us denote by V 1 the subspace from the rght-hand member of (1). If v V 1 then partcularly v ( r =1 A k )A (k +1) C, hence A v ( r =1 A k )C, k N, {1, 2,..., r}. It follows that A v V 1, {1, 2,..., r}.e. V 1 s (A 1,..., A r )-nvarant. We can wrte by (1) V 1 = C k r =0 ( r =1 A k )C where (k 1,..., k r ) (0,..., 0), hence V 1 s ncluded n C. Now, let V be any (A 1,..., A r )-nvarant subspace ncluded n C. Then, for any v V, ( r =1 Ak )v V C, hence v ( r =1 A k )C, k 0, {1, 2,..., r}, whch mples v V 1. Therefore V V 1,.e. V 1 s the maxmal such subspace. Proposton 2.2. The maxmal (A 1,..., A r )-nvarant subspace ncluded n C s maxi(a 1,..., A r ; C) = n 1 n 1 ( k r =0 =1 A k )C. (2) Proof. Let us denote by V 2 the subspace from the rght-hand member of (2). Obvously, by Proposton 2.1, V 1 V 2, where V 1 = maxi(a 1,..., A r ; C). Now, for any v V 2, ( r =1 Ak )v C, k N, 0 k n 1, {1, 2,..., r}. Let p (s) = det(si A ) = s n +a n 1, s n 1 ++a 1, s+a 0, be the characterstc polynomal of the matrx A, {1, 2,..., r}. By Hamlton-Cayley Theorem, p (A ) = 0 n, hence A n = a n 1, A n 1 a 1, A a 0, I n. (3) n 1 Then, for any vector v V 2, A n v = a l, A l v. Snce A are commutatve matrces, we can premultply ths equalty by ( r =1 A k ) and we obtan ( r hence ( r =1 A k )An v C snce ( r =1 l=0 A k )Al Smlarly, by postmultplyng (3) by ( r =1 v C and fnally that ( r that ( r =1 A k )An+t =1 Ak follows that V 2 V 1, hence V 2 = V 1 = maxi(a 1,..., A r ; C). =1 A k n 1 )An v = a l, ( l=0 =1 v C for 0 l n 1 and C s a subspace. A k )At v, t = 1, 2,..., one obtans recurrently )v C, k 0, hence v V 1. It A k )Al v, Algorthm 2.1. Stage 1. Determne the sequence of subspaces (S 1,0,...,0,0) 0 1 n of the space X = K n : S 0,0,...,0,0 = C; (4) S 1,0,...,0,0 = C A 1 1 S 1 1,0,...,0,0, 1 = 1,..., n; (5)

4 16 Valeru Prepelţă, Tberu Vaslache Stage 2. Determne 0 1, the frst ndex n {0, 1,..., n 1} whch verfes S ,0,...,0,0 = S 0 1,0,...,0,0. (6) % S 0 1,0,...,0,0 s the maxmal A 1 -nvarant subspace whch s ncluded n C. If 0 1 = n 1, then maxi(a 1,..., A r ; C) = {0} K n. STOP If 0 1 < n 1, put := 2 and GO TO Stage 3. Stage 3. Determne by (7) the sequence of subspaces (S 0 1, 0 2,...,0,,0,...,0) 0 n of the space X = K n : for = 1, 2,..., n, % S 0 1, 0 2,...,0,0,0,...,0 s the maxmal (A 1,..., A )-nvarant subspace whch s ncluded n C S 0 1, 0 2,...,0,,0,...,0 = S 0 1, 0 2,...,0,,0,...,0 A 1 S 0 1, 0 2,...,0,,0,...,0. (7) Stage 4. Determne 0, the frst ndex n {0, 1,..., n 1} whch verfes S 0 1, 0 2,...,0,0 +1,0,...,0 = S 0 1, 0 2,...,0,0,0,...,0. (8) If 0 = n 1 then maxi(a 1,..., A r ; C) = {0} K n. STOP If 0 < n 1 then GO TO Stage 5. Stage 5. If < r then put := + 1 and GO TO Stage 3. If = r, then maxi(a 1,..., A r ; C) = S 0 1, 0 2,...,0,0,...,0 r. STOP Proof. We consder the rd chan of subspaces S 1, 2,..., r = 1 2 k 2=0 r k r=0 1 A k2 2 A k r r C, (9) {0, 1,..., n}, {0, 1,..., r}. Obvously, for l, {0, 1,..., r}, S 1, 2,..., r S l1,l 2,...,l r. (10) By (9) and Proposton 2.1 we obtan S 1, 2,..., r maxi(a 1,..., A r ; C), 0, {1, 2,..., r} and t follows from Proposton 2.2 that S n 1,n 1,...,n 1 = maxi(a 1,..., A r ; C). From (4) and (9) one obtans S 0,0,...,0,0 = A 0 1 A 0 2 A 0 r C = C = S 0,0,...,0,0. Let us assume that S 1 1,0,...,0,0 = S 1 1,0,...,0,0 for some 1 {1, 2,..., n}. Applyng (9), (5) and the change of the ndex k 1 1 = k, we get S 1,0,...,0,0 = 1 A k 1 1 C = A 0 1 C A k 1 =1 A (k 1 1) 1 C = C A k=0 A k 1 C = C A 1 1 S 1 1,0,...,0,0 = S 1,0,...,0,0, hence we obtaned by nducton and by (10) the followng relatons: S 1,0,...,0 = S 1,0,...,0, 1 {1, 2,..., n}, (11) S 1,0,...,0 S 1+1,0,...,0, 1 {0, 1,..., n 1}. (12) Usng Hamlton-Cayley Theorem as n the proof of Proposton 2.2, one obtans S n,0,...,0 = S n 1,0,...,0, hence by (11), S n,0,...,0 = S n 1,0,...,0,.e. 0 from (6) verfes 1

5 Maxmal Invarant Subspaces and Observablty of Multdmensonal Systems. Part 2: the Algorthm 17 0 n 1. Let us consder the chan of subspaces K n S 0,0,...,0,0 S 1,0,...,0,0... S 1,0,...,0,0... S n 1,0,...,0,0 = S n,0,...,0,0. Snce S 0,0,...,0,0 = C s a proper subspace of K n, dm S 0,0,...,0,0 n 1. If 0 = n 1 s the frst ndex whch verfes (6), t follows that n 1 dm S 0,0,...,0,0 > dm S 1,0,...,0,0 >... > dm S n 1,0,...,0,0 0, hence dm S n 1,0,...,0,0 = 0 and we have {0} = S n 1,0,...,0,0 = S n 1,0,...,0,0 maxi(a 1,..., A r ; C) {0}. Therefore maxi(a 1,..., A r ; C) = {0}, whch proves the nstructon n Stage 2. If 0 1 < n 1, one obtans by (5) and (6) S ,0,...,0,0 = C A 1 1 S ,0,...,0,0 = C A 1 1 S 0 1,0,...,0,0 = S ,0,...,0,0 = S 0 1,0,...,0,0. Let us assume that S 1,0,...,0,0 = S 0 1,0,...,0,0 for some 1 { ,..., n}. Then, applyng agan (5) and (6), we get S 1 +1,0,...,0,0 = C A 1 1 S 1,0,...,0,0 = C A 1 1 S 0 1,0,...,0,0 = S ,0,...,0,0 = S 0 1,0,...,0,0, hence we proved by nducton that S 1,0,...,0,0 = S 0 1,0,...,0,0, 1 { ,..., n}. Now, let us assume that S 0 1,..., 0,,0,...,0 = S 0 1,..., 0,,0,...,0 for some {2,..., r} and {1,..., n}. k=0 ( Consder some subspaces V k K n, k = 0, 1,...,, N. We have 1 V k = V 0 ( k=1 1 V k ) V = [V 0 ( k=1 1 V k )] [( Therefore, by replacng V k by A k C, we have k =0 k =0 Usng ths equalty and (6), we get k =0 S 0 1,..., 0,,0...,0 = 1 A k k=1 1 C = ( C) ( k = k =0 A k C) ( 1 V k ) V ] = ( k =1 C). 1 A k k =0 k =0 k=0 V k ) ( C = 1 A k k =1 V k ). k=1 A k C), whch becomes by the backward movement of A 1 and the change of the ndex k 1 = k n the last term 0 1 ( 0 k =0 1 A k 1 k =0 0 1 C) A 1 ( 0 k =0 1 A k A (k) k =1 C) =

6 18 Valeru Prepelţă, Tberu Vaslache 0 1 =( 0 k =0 1 A k 1 k =0 0 1 C) A 1 ( 0 k =0 whch s equal by (9), by the nducton assumpton and by (7) to 1 A k S 0 1,..., 0,,0...,0 A 1 S 1v,..., 0,,0...,0 = = S 0 1,..., 0,,0...,0 A 1 S 0 1,...,, 0,0...,0 = S 0 1,..., 0,,0...,0, hence we proved by nducton that therefore {1,..., n} 1 k=0 A k C), S 0 1,..., 0,,0,...,0 = S 0 1,..., 0,,0,...,0, {1,..., r}, {1,..., n}, (13) S 0 1,..., 0,...,0 r = S 0 1,...,0,...,0 r. (14) By (9),(13) and (12),we obtan, for 0 1,..., 0 determned n Stage 4 and S 0 1,..., 0,,0,...,0 = k =0 k =0 1 A k A k C (15) and S 0 1,..., 0,,0,...,0 S 0 1,..., 0,+1,0,...,0. By (15) and by applyng Hamlton-Cayley Theorem to the matrx A, one obtans S 0 1,..., 0,n,0,...,0 = S 0 1,..., 0,n 1,0,...,0. Let us consder the chan of subspaces K n S 0,0,...,0,0 S 0 1,..., 0,0,0,...,0... S 0 1,..., 0,1,0,...,0... S 0 1,..., 0,n 1,0,...,0 = S 0 1,..., 0,n,0,...,0. that If 0 = n 1 s the frst ndex whch verfes (8), snce dm S 0,0,...,0,0 n 1, t follows n 1 dm S 0 1,..., 0,0,0,...,0 > dm S 0 1,..., 0,1,0,...,0 >... > dm S 0 1,..., 0,n 1,0,...,0 0, whch mples dm S 0 1,..., 0,n 1,0,...,0 = 0, hence S 0 1,..., 0,n 1,0,...,0 = {0}. By Proposton 2.2 and (15), maxi(a 1,...,A r ;C) S 0 1,..., 0,n 1,0,...,0, hence maxi(a 1,...,A r ;C)= {0}, whch proves the nstructon from Stage 4. Consder the case 0 < n 1 (condton whch ncludes 0 k < n 1, k {1,..., 1}). We have by (8) S 0 1,..., 0,0 +1,0,...,0 = S 0 1,..., 0,0,0,...,0. Let us assume that S 0 1,..., 0,,0,...,0 = S 0 1,..., 0,0,0,...,0 for some 0. Then S 0 1,..., 0,+1,0,...,0 = S 0 1,..., 0,,0,...,0 A 1 S 0 1,..., 0,,0,...,0 = S 0 1,..., 0,0,0,...,0 A 1 S 0 1,..., 0,0,0,...,0 = S 0 1,..., 0,0 +1,0,...,0 = S 0 1,..., 0,0,0,...,0. We proved by nducton S 0 1,..., 0,,0,...,0 = S 0 1,..., 0,0,0,...,0, { 0 + 1,..., n 1}. (16)

7 Maxmal Invarant Subspaces and Observablty of Multdmensonal Systems. Part 2: the Algorthm 19 By (13) and (16) we obtan S 0 1,..., 0,,0,...,0 = S 0 1,..., 0,0,0,...,0, { 0 + 1,..., n 1}. (17) For = 1 we proved that f 0 1 < n 1, then S 1,0,...,0 = S 0 1,0,...,0, 1 { ,..., n 1}. Usng (11), we get S 1,0,...,0 = S 0 1,0,...,0, 1 { ,..., n}. Let us now assume that S 1,...,,0,0,...,0 = S 0 1,..., 0,0,0,...,0, k { 0 k + 1,..., n 1}, k {1, 2,..., 1},.e. (see (9)) Then k =0 A k 1 1 k =0 k =0 1 A k C = 1 A k C= 0 1 k =0 A k and usng the commutatvty of the matrces A we have 1.e. agan by (9) k =0 k =0 1 A k C= k = A k C. k =0 k =0 S 1,...,,,0,...,0 = S 0 1,..., 0,,0,...,0, and usng (17) we obtan by nducton, for any {1, 2,..., r} 0 k =0 1 A k C 1 A k C S 1,...,,,0,...,0 = S 0 1,..., 0,0,0,...,0, k { 0 k + 1,..., n 1}, k {1, 2,..., }, (18) partcularly S 1,...,,..., r = S 0 1,..., 0,,...,0 r, k { 0 k + 1,..., n 1}, k {1, 2,..., r}. (19) It follows by Proposton 2.2 and (14) that maxi(a 1,..., A r ; C) = S n 1,...,n 1,...,n 1 = S 0 1,..., 0,,...,0 r, (20) whch proves the fnal statement from Stage 5. By (9) and (20) we obtan Proposton 2.3. The maxmal (A 1,..., A r )-nvarant subspace ncluded n C s maxi(a 1,..., A r ; C) = r k r =0 where 0 1,..., 0 r are the numbers determned n stages 2 and 4. 1 A k r r C,

8 20 Valeru Prepelţă, Tberu Vaslache 3. Matlab program The Matlab program presented below and based upon the algorthm above calculates the dmenson and an orthonormal bass of the maxmal nvarant subspace. The nstructons make use of the m-functons nts, nvt and ma ncluded n the Geometrc Approach toolbox publshed by G. Marro and G. Basle at ths GA toolbox works wth Matlab and Control System Toolbox. More precsely, gven the matrces A 1, A 2... A r that commute and the matrx C, the next commands wll compute and dsplay the dmenson of a bass and an orthonormal one n the subspace S = maxi(a 1, A 2,... A r ; C), where C = ImC. The matrces are loaded from the m-fle GetAC.m, where A 1, A 2... A r are stored n an 1 r dmensonal cell array A as A{1},...A{r}. % begn m-fle % nts(x,y)=an orthonormal bass for Im(X) ntersected wth ImYB) % nvt(x,y)=an orthonormal bass for the nverse mage of Y through X % ma(z) = an othhonormal bass n the subspace generated by Z Get_A_C; % A s a 1 x n cell array, contanng A{1},... A{r} [ ~, r] = sze(a); S = ma(c); [n, dmmax] = sze(s); % wll be the dmenson of the maxmal nvarant subspace ndex = zeros(1, r); % the ndex of the calculated subspace for = 1:r % loop for ndex poston for = 1:n-1 % loop for ndex value S = nts(s, nvt(a{},s)); [~, m1] = sze(s); ndex() = ; f (m1 == dmmax) break; else dmmax = m1; end end f ((dmmax == 1) && (norm(s, 2) == 0)) dmmax = 0; break; end end dsp([ The dmenson of the maxmal nvarant subspace s ]) dsp([ num2str(dmmax)]) dsp( and an orthonormal bass of ths subspace s: ) dsp(s) % end m-fle

9 Maxmal Invarant Subspaces and Observablty of Multdmensonal Systems. Part 2: the Algorthm 21 For example, gven the commutng matrces A 1 = , A 2 = A 3 = and C = , the m-fle GetAC could be A = cell(1, 3); A{1} = [1,2,1,3;-2,-2,2,-2;1,2,1,1;2,0,-2,0]; A{2} = [1,2,-1,5;-6,-4,6,-6;-1,2,1,-1;6,0,-6,2]; A{3} = [1,3,3,2;2,-2,-2,0;3,3,1,4;-2,0,2,-2]; C = [ ;0 1 0 ;0 0 1 ; ]; and the above Matlab program wll gve the answers: The dmenson of the maxmal nvarant subspace s 2 and an orthonormal bass of ths subspace s: , 4. Applcaton to the observablty of a class of dscrete-tme rd systems In ths secton we wll show how Algorthm 2.1 can be appled to problems concernng the concept of observablty of a class of multdmensonal systems. We shall use the followng notatons: r := {1, 2,..., r} where r N. A functon x(t 1,..., t r ) s denoted by x(t), where t = (t 1,..., t r ), and varables. t Z + are the dscrete-tme For a subset δ = { 1,..., l } of r, we consder the notatons δ := l, δ := r \ δ and := 0; for r, ĩ := r{}. The notaton δ r means that δ s or δ s a subset of r and δ r. For δ = { 1,..., l }, the shft operator σ δ s defned by σ δ x(t) = x(t + e δ ) where e δ = e e l, e = (0,..., 0, 1, 0,..., 0) Z }{{} r ; when δ = r we denote σ δ = σ, hence σx(t 1, t 2,..., t r ) = x(t 1 + 1, t 2 + 1,..., t r + 1). Defnton 4.1. An rd dscrete-tme lnear system s an ensemble Σ = (A 1,..., A r ; B; C; D) where A, r are commutng n n matrces over a feld K and B, C, D are respectvely

10 22 Valeru Prepelţă, Tberu Vaslache n m, p n and p m matrces over K. The K-spaces X = K n, U = K m, Y = K p are called respectvely the state space, nput space and output space and the tme set s T = N r. The followng equatons are called respectvely the state equaton and the output equaton: σx(t) = δ r ( 1) r δ 1 A σ δ x(t) + Bu(t), (21) δ y(t) = Cx(t) + Du(t), (22) where x(t) = x(t 1,..., t r ) X s the state, u(t) U s the nput and y(t) Y s the output of the system Σ at the moment t T. For any set δ = { 1,..., l } r and for t Z +, δ, we use the notaton x(t δ, 0 δ) := x(0,..., 0, t 1, 0,..., 0, t l, 0,..., 0). Defnton 4.2. The vector x 0 K n s called an ntal state of the system Σ f ( ) x(t δ, t 0 δ) = A x 0 (23) for any δ r; equaltes (23) are called ntal condtons of Σ. δ By [8, Proposton 2.2], we have Theorem 4.1. The output of the system Σ at the moment t, determned by the ntal state x 0 and the output u : T U s (wth l = (l 1,..., l r )): t 1 1 t r 1 y(t) = C x C =1 A t l 1 =0 l r =0 =1 A t l 1 Bu(l) + Du(t). (24) Defnton 4.3. A state x K n s sad to be unobservable f, for any nput u(t), the ntal states x 0 = x and x 0 = 0 produce the same output y(t), t T. Proposton 4.1. The state x K n s unobservable f and only f C =1 A t x = 0, t N, r. (25) Proof. We denote by y x (t) and y 0 (t) the outputs produced by the ntal state x 0 = x and x 0 = 0 respectvely, for an arbtrary nput u(t). We obtan by (24) from y x (t) = y 0 (t) that the state x s unobservable f and only f y x (t) y 0 (t) = 0 t T whch s equvalent to (25).

11 Maxmal Invarant Subspaces and Observablty of Multdmensonal Systems. Part 2: the Algorthm 23 In the sequel we wll consder the system Σ reduced to the ensemble Σ = (A 1,..., A r ; C) whch s nvolved n formulas concernng observablty. Usng Hamlton-Cayley Theorem as n the proof of Proposton 2.2, we deduce from (25) the followng result. Proposton 4.2. The state x K n s unobservable f and only f C =1 A t x = 0, t N, t n 1, r. (26) Let us denote by X uo the set of the unobservable states of Σ and by C the subspace C = KerC. The next theorem gves the geometrc characterzaton of the set of unobservable states of Σ. Theorem 4.2. X uo s the maxmal (A 1,..., A r ; C)-nvarant subspace of K n. Proof. Let x be an unobservable state of Σ. Obvously, one obtans from (25), for t = 0, r, that Cx = 0, hence X uo KerC. For arbtrary r and t N, r one obtans from (25): 0=C( A t )At +1 x= C( =1 A t )A x, hence A x X uo, r,.e. X uo s (A 1,..., A r ; C)-nvarant. Now, consder an arbtrary (A 1,..., A r ; C)-nvarant subspace V of K n and let v be an element of V. Snce V s (A 1,..., A r )-nvarant and t s ncluded n KerC one obtans ( A t )v V and C( A t )v = 0, t 0, r. By (25), v X uo, hence V X uo,.e. =1 =1 X uo s the maxmal such space. =1 Defnton 4.4. The system Σ s sad to be completely observable f there s no unobservable state x 0. Therefore, the system Σ s completely observable f and only f X uo = {0}. From Theorem 4.2 we obtan Theorem 4.3. The system Σ s completely observable f and only f {0} s the maxmal (A 1,..., A r ; C)-nvarant subspace of K n. By Theorem 4.2 we can use Algorthm 2.1 to determne the subspace X uo of the unobservable states of a multdmensonal system Σ. If one gets n Stage 2 the value 0 1 = n 1 or n Stage 4 0 = n 1, t follows that X uo = {0}, hence, by Theorem 4.3, Algorthm 2.1 can be used to check f the system Σ s completely observable.

12 24 Valeru Prepelţă, Tberu Vaslache To ths am we modfy Algorthm 2.1 by replacng C by KerC n Stage 1 (see (4) and (5)) and n Stages 2 and 4. We also replace maxi(a 1,..., A r ; C) by X uo n Stages 2,4 and 5. We wrte The system s completely observable n Stages 2 and 4 and The system s not completely observable n Stage Conclusons An algorthm s proposed and ts proof s gven, n the lnes of the Geometrc Approach. Ths algorthm determnes the maxmal nvarant subspace wth respect to a fnte number of commutng matrces and whch s ncluded n a gven subspace. Ths algorthm can be used n the study of the observablty of multdmensonal (nd) lnear systems, especally to fnd the canoncal form of the unobservable systems and (combned wth the dual algorthm for the controllable subspace), the Kalman canoncal decomposton of a multdmensonal system. R E F E R E N C E S [1] S. Attas, Introducton d une classe de systèmes lnéares reccurents à deux ndces, Comptes Rendus Acad. Sc. Pars sére A, 277 (1973), [2] G. Basle, G. Marro, Controlled and condtoned nvarant subspaces n lnear system theory, J. Optmz. Th. Applc., 3 (1969), No. 5, [3] G. Basle and G. Marro, Controlled and Condtoned Invarants n Lnear System Theory, Upper Saddle Rver, NJy: Prentce-Hall, [4] E. Fornasn, G. Marchesn, State space realzaton theory of two-dmensonal flters, IEEE Trans. Aut. Control, AC-21 (1976), [5] G. Marro, Teora de sstem e del controlo, Zanchell, Bologna, 1989 [6] V. Prepelţă, Observablty of a model of (q,r)-d contnuous-dscrete systems, Proceedngs of the 4th Internatonal Conference on Dynamcal Systems and Control (CONTROL 08), Corfu Island, Greece, October 26-28, (2008), [7] V. Prepelţă, T. Vaslache, Mona Dorofte, Geometrc approach to a class of multdmensonal hybrd systems, Balkan Journal of Geometry and Its Applcatons, 17 (2012), No. 2, [8] V. Prepelţă, T. Vaslache, Maxmal Invarant Subspaces and Observablty of Multdmensonal Systems, U.P.B. Sc. Bull., Seres A, 77 (2015), No. 2, [9] R. P. Roesser, A dscrete state-space model for lnear mage processng, IEEE Trans. Aut. Control, AC-20 (1975), [10] W. M. Wonham, A. S. Morse, Decouplng and pole assgnment n lnear multvarable systems: A geometrc approach, SIAM J. Control, 8 (1970), 1-18.