A fast algorithm for subspae state-spae
system identiation via exploitation of the
displaement struture
Niola Mastronardi a d 2, Daniel Kressner b 3, Vasile Sima a e 4,
Paul Van Dooren 3 5, and Sabine Van Huel a 1 2 3
;
;
;
; ;
; ;
; ; ;
Department of Eletrial Engineering, ESAT-SISTA/COSIC, Katholieke
Universiteit Leuven, Kardinaal Merierlaan 94, 3001 Heverlee, Belgium
b
Department of Mathematis, University of Chemnitz, Chemnitz, Germany
Department of Mathematial Engineering, Universite Catholique de Louvain,
Avenue Georges Lemaitre 4 B-1348 Louvain-la-Neuve, Belgium
d
Dipartimento di Matematia, Universita della Basiliata, via N. Sauro 85, 85100
Potenza, Italy
e
National Institute for Researh & Development in Informatis, Bd. Maresal
Averesu, Nr. 8{10, 71316 Buharest 1, Romania
a
Abstrat
Two reent approahes [15,16℄ in subspae identiation problems require the omputation of the R fator of the QR fatorization of a blok{Hankel matrix H , whih,
in general has a huge number of rows. Sine the data are perturbed by noise, the
involved matrix H is, in general, full rank. It is well known that, from a theoretial point of view, the R fator of the QR fatorization of H is equivalent to the
Cholesky fator of the orrelation matrix H T H , apart from a multipliation by a
sign matrix. In [12℄ a fast Cholesky fatorization of the orrelation matrix, exploiting the blok{Hankel struture of H , is desribed. In this paper we onsider a fast
algorithm to ompute the R fator based on the generalized Shur algorithm. The
proposed algorithm allows to handle the rank{deient ase.
Key words: generalized Shur algorithm, Hankel and blok{Hankel matries,
subspae identiation, QR deomposition, singular value deomposition.
Preprint submitted to Elsevier Preprint
1 Introdution
Subspae based system identiation has beome very popular in the last
deade [4℄. The suess of this state{spae identiation approah is mainly
due to the fat that it relies on a simple matrix deomposition for whih
reliable numerial algorithms are available. Its major drawbak, on the other
hand, is that large \data" matries are involved and that this may lead to
high omputing and storage osts. We now briey reall the basi formulation
of the problem. Let uk and yk be the m{dimensional input vetor and the
l{dimensional output vetor, respetively, of the linear time{invariant state{
spae model
xk+1 = Axk + Buk + wk ;
yk = Cxk + Duk + vk ;
where xk is the n{dimensional state vetor at time k, fwk g and fvk g are state
and output disturbanes or noise sequenes, and A, B , C and D are unknown
real matries of appropriate dimensions.
For non{sequential data proessing, one hooses N 2(m + l)s and onstruts
the N 2(m + l)s matrix H = [U Ts;N Y Ts;N ℄, where U s;N and Y s;N are blok{
2
2
2
2
This author is a Senior Researh Assoiate with the F.W.O. (Fund for Sienti
Researh-Flanders).
2
This work is supported by UE Programme \Training and Mobility of Researhers"
projet (ontrat ERBFMRXCT970160) entitled \Advaned Signal Proessing of
Medial Magneti Resonane Imaging and Spetrosopy".
3
This work is supported by the Belgian Programme on Interuniversity Poles of Attration (IUAP-4/2 & 24), initiated by the Belgian State, Prime Minister's OÆe for
Siene, and by a Conerted Researh Ation (GOA) projet of the Flemish Community, entitled \Mathematial Engineering for Information and Communiations
Systems Tehnology".
4
This work is supported by the European Community BRITE-EURAM III Themati Networks Programme NICONET (projet BRRT{CT97-5040).
5
This work is partly supported by the National Siene Foundation ontrat CCR97-96315.
1
2
Hankel matries dened in terms of the input and output data, respetively :
2
U2s;N
=
6
6
6
6
6
6
6
6
6
6
6
6
4
2
Y2s;N
=
6
6
6
6
6
6
6
6
6
6
6
6
4
u1 u2
u2 u3
u3 u4
...
...
u3 : : : uN
u4 : : : uN +1
u5 : : : uN +2
...
...
...
7
7
7
7
7
7
7;
7
7
7
7
7
5
...
u2s u2s+1 u2s+2 : : : uN +2s
y1 y2
y2 y3
y3 y4
3
1
3
y3 : : : yN
y4 : : : yN +1
y5 : : : yN +2
...
7
7
7
7
7
7
7:
7
7
7
7
7
5
...
y2s y2s+1 y2s+2 : : : yN +2s
1
Then the R fator of a QR fatorization H = QR is used for data ompression.
In [12℄ a fast Cholesky fatorization of the orrelation matrix, exploiting the
blok{Hankel struture of H , is desribed. In this paper we onsider a fast
algorithm to ompute the R fator of the QR fatorization of H based on the
generalized Shur algorithm, exploiting its displaement struture. The paper
is organized as follows. In x2 the generalized Shur algorithm to ompute the
Cholesky fator of a symmetri positive denite matrix is desribed and in x3
this algorithm is applied to the matrix H . The rank{deient ase is desribed
in x4 and some numerial experiments are reported in x5.
2 The Shur algorithm for positive semi-denite matries
2.1 The Cholesky fator and generator of A
We summarize here the key properties of the generalized Shur algorithm to
ompute the Cholesky fator of a (symmetri) positive semi-denite (psd)
matrix, whih will be used in the next setion. More details an be found in
[7,8℄. Let A be a psd matrix of order n, then we dene its displaement
rA = A Z T AZ;
using a generalized shift matrix Z of the same dimension. Here we only require
the shift matrix Z to be stritly upper triangular (and hene nilpotent) and
3
we speialize later on to a partiular hoie of Z . We all the rank of rA the
displaement rank of A and we assume that it is signiantly smaller than
n. Let the symmetri matrix rA have p positive eigenvalues and q =: p
negative eigenvalues then it has a fatorization
2
0
I
rA =: GT G; =: 64 p
0 Iq
3
7
5;
2
G =: 64
3
Gp 7
5:
Gq
(1)
The matrix G is alled the generator of A and sine Z is nilpotent one an
reonstrut A via the formula
A=
nX1
i=0
(Z i)T GT GZ i :
The generator and intermediate results derived from transformations of the
generator, allow to reonstrut the Cholesky fator of the psd matrix A :
3
2
6 r1;1 r1;2 : : : rn;n 7
7
6
7
6
r
6
2;2 : : : r2;n 7
7
6
A = RT R; R = 6
. . . ...
6
6
4
rn;n
7:
7
7
5
Notie that if A has rank r < n then so will the fator R whih will have its
last n r rows equal to zero :
2
A=
RT
R; R =
6 r1;1
6
6
6
6
4
3
: : : : : : rn;n 7
...
...
rr;r : :rr;n
7
7
7:
7
5
If the leading r r prinipal submatrix of A is nonsingular then ri;i; i = 1; : : : ; r
are all nonzero. Otherwise the \prole" of the trapezoidal matrix R indents to
the right eah time the nullity of the i i prinipal submatrix of A inreases.
In our appliation, A is a produt of the type H T H whih learly is a psd
matrix. Its Cholesky fator R is, up to a sign matrix D = diag(1; : : : ; 1),
also the RH fator of the QH RH deomposition of H : RH = DR. Hene both
the problem of omputing the RH fator of the QR fatorization of H and that
of omputing the Cholesky fator of H T H are equivalent. We disuss now the
omputation of the Cholesky fator R of A starting from the generator of rA.
4
One easily shows that the generator G is not unique. We say that the generator
G~ is proper if its rst olumn is zero exept possibly its leading element. The
following theorem holds for proper generators [7℄.
Theorem 1 Let
3
2
a11 a12 7
A = 64
5
a21 A22
be a positive semi-denite matrix with proper generator
2
3
2
6 g~1;1 g~1;2 g~1;n 7
6
6
7
6
6
7
6
6 0 g~2;2 g~2;n 7 : 6
~G = 66 . .
7=6
.. 77 66
..
6 ..
. 7 6
6
4
5
4
0 g~;2 g~;n
G~ 1
G~ 2
3
7
7
7
7
7;
7
7
7
5
then G~ 1 is the rst row of the Cholesky fator R of A. Furthermore the gen^ where
erator matrix for the Shur omplement A^ = A G~ T1 G~ 1 is given by G;
2
60
A^ =
6
6
6
6
6
60
6
4
0
A22
a21 a a12
1
11
3
2
7
7
7
7
7;
7
7
7
5
6
6
6
6
6
6
6
6
4
G^ =
G~ 1 Z
G~ 2
3
7
7
7
7
7:
7
7
7
5
We observe that the rst olumn of G^ is zero, whih needs to be the ase sine
the rst olumn and row of A^ are zero.
The generalized Shur algorithm just onsists of a reursive use of this Theorem : via a transformation (dened below) the generator G of the urrent
~ This yields the urrent row of the Cholesky
matrix A is put in proper form G:
fator and the generator of the Shur omplement is trivially obtained from
a shift Z applied to the rst row of the generator. We refer to [7℄ for more
details. The omplexity of this algorithm is that of the transformation sine
the shift Z does not imply any operations. In the next setion we desribe
briey the onstrution of .
5
2.2 Redution of the generator to proper form
The rst row of the Cholesky fator R of A is thus obtained from a proper
generator G of A. Reduing a non-proper generator of A to a proper generator
~ is obtained by applying a transformation to the generator G. In order
G;
not to hange the produt GT G it suÆes to hoose to be -unitary, i.e. :
T = ;
sine then
G~ T G~ = (G)T (G) = GT G:
Typially the matrix is onstruted as follows :
2
=:
6
6
6
6 Ip
6
6
6
6
4
3 2
7 6
7 6
7 6
7 6
7:6
7 6
7 6
7 6
5 6
4
1
Iq
3
Hp
1
7
7
7
7
7
7:
7
7
7
Hq 5
(2)
The bloks Hp and Hq of the seond fator are p p and q q Householder
transformations reduing the rst olumn of G as follows
2
2
6 Hp
4
32
Hq
76
54
3
Gp
Gq
7
5
=
6 x11
6
6
6
6
6
6
6
6
6
6
6
6
6
6 y11
6
6
6
6
6
6
6
6
4
x12
0
... X
0
22
y12
0
... Y
0
6
22
3
7
7
7
7
7
7
7
7
7
7
7
7
7:
7
7
7
7
7
7
7
7
7
7
5
(3)
The rst fator :only transforms the rows ontaining x and y and eliminates
y provided = y =x is smaller than one in modulus :
11
11
11
11
11
3
3 2
2
3
2
~11 x~ 12 7
6x
6 7 6 x11 x12 7
5:
5=4
5:4
4
0 y~12
y11 y12
This 2 2 transformation is onstruted from =: 1=p1 , =: =p1 and is alled aphyperboli rotation sine it satises = 1 [1℄. Also note
that x~ = x 1 . When it is implemented in fatored form :
2
2
11
2
2
2
11
2
6
4
1
0
p
1 2
32
76
56
4
p
32
3
0 77 6 1 7
5;
54
0
1
0 1
1
1
2
one shows that the generalized Shur algorithm is bakward stable and that
it has the same omplexity as the unfatored implementation [13℄.
It follows already from (3) that the (1; 1) element of the psd matrix A equals
a =x
y 0. Therefore if a 6= 0 the above transformation an be
performed. On the other hand, if a = 0 then the whole row a must be
zero sine otherwise A would not be psd. Sine a = x x y y this also
implies that [x ; x ℄ = [y ; y ℄ and that both these rows an just be
deleted from the generator [5℄. In other words, if a = 0 a simpliation an
be performed to the urrent generator G~ . The omplexity of the redution of
G to proper form G~ is essentially that of the Householder transformations Hp
and Hq whih osts 4(p + q)n ops. If r = rank A steps are performed, this
algorithm thus requires a total of 4r(p + q)n ops. We point out that this
is an overestimate sine the number of nonzero olumns n of the generator
dereases at eah step and that potentially the number of rows p + q may
derease as well.
2
11
11
2
11
11
11
12
12
11
12
11
11
12
11
12
12
11
3 Fast omputation of the R fator of the QR fatorization of H
We show here how to ompute the gererator G of A = H T H where H 2
R N; m l s is the blok Hankel matrix desribed in the rst setion with bloks
2(
+ )
7
of sizes 1 m and 1 l :
2
uT
6 1
6
6 T
6 u2
6
6
6
6
4
uTN
uT
2
uT2s
:::
yT
...
...
... ...
H= . . .
.. . . . .
1
yT
2
:::
2
1
7
7
7
7
7:
7
7
7
5
...
...
... ...
... . . . . . .
yT
: : : : : : uTN +2s
3
y2Ts
yNT : : : : : : yNT +2s
1
The shift matries used in this ontext are the matries
6 0m Im
6
6
0m
6
:
Zm = 66
6
6
4
...
. . . Im
0m
3
2
3
2
6 0l Il
7
6
7
6
7
0l
6
7
:
7 ; Zl = 6
6
7
6
7
6
4
7
5
7
. . . 777
:
. . . Il 777 ; Z = Zm Zl : (4)
7
5
0l
The following theorem then gives a ontrution of a generator for A.
Theorem 2 Given the QR fatorization of the rst blok olumns :
2
T
6 u1
6
6 T
6 u2
6
6 ..
6 .
6
4
uTN
2
3
3
y1T 7 6 q1T 7
7 6
7
y2T 777 666 q2T 777
.. 77 = 66 .. 77 R1
. 7 6 . 7
5 4
5
yNT
qNT
(5)
where R1 an be assumed upper trapezoidal of row rank k m + l and qi 2 R k ;
dene the produt
Cu;1 : : : Cu;2s Cy;1 : : : Cy;2s
= q : : : qN H:
1
(6)
Then a generator G for H T H is given by
= Ik Ik
G = Gu Gy ;
+1
8
+1
(7)
where
2
Gu =
6 Cu;1
6
6
6
6
6
6
6
4
0
0
0
2
3
3
Cu;2 : : : Cu;2s 7
6 Cy;1 Cy;2 : : : Cy;2s 7
6
7
7
6 0 yT
7
T
uTN +1 : : : uTN +2s 1 777
:
:
:
y
6
N +1
N +2s 1 7
6
7:
;
G
=
y
6
7
7
6
7
7
Cu;2 : : : Cu;2s 7
6 0 Cy;2 : : : Cy;2s 7
4
5
5
uT1 : : : uT2s 1
0 y1T : : : y2Ts 1
Proof :
In order to prove the result we onsider the displaement matrix rH T H :
2
T
6 U2s;N U2s;N
4
Y2s;N U2Ts;N
ZmT U2s;N U2Ts;N Zm U2s;N Y2Ts;N
ZlT Y2s;N U2Ts;N Zm Y2s;N Y2Ts;N
3
ZmT U2s;N Y2Ts;N Zl 7
5
ZlT Y2s;N Y2Ts;N Zl
(8)
whih ought to be equal to
3
2
T
T
6 Gu Gu Gu Gy 7
5:
4
GTy Gu GTy Gy
(9)
It follows from (5,6) that R = [Cu; Cy; ℄ and hene
1
1
1
[Cu; Cy; ℄ Cu; : : : Cu; s Cy; : : : Cy; s =
1
1
T
1
2
1
2
RT
1
q1 : : : qN H;
whih are the rst blok rows of the sub-bloks of (8). This thus veries the
rst blok rows and blok olumns of the equality between (8) and (9). The
rest easily follows from the blok Hankel struture of H .
Note that if the rst blok olumns of H in (5) have full rank then R is square
invertible and k = m + l. If moreover the whole matrix H has full olumn
rank, then the generalized Shur algorithm will not enounter any singularities.
But sine the low rank ase is of partiular interest here, singularities in the
generalized Shur algorithm will be enountered and hene lead to a lower
omplexity of the algorithm.
The above theorem also shows that the displaement rank of H T H is at most
2(k + 1) 2(m + l + 1), with the same number of positive and negative
generators. Hene the generalized Shur algorithm to ompute the R fator
requires about (8Nrk) ops. To ompute the generator G of H T H , the QR
1
9
Table 1
Numerial results for Example 1
RM
RS
bakward error RM
bakward error RS
numerial rank
# ops # ops
31660
1:51 10
9489
5:20 10
16
5
15
fatorization (5) requires (6N (m + l) ) ops and the produt (6) requires less
than (4Nk(m + l)s) ops. We reall that k (m + l) and r 2(m + l)s
but that equality is obtained when no rank deieny is deteted. The most
time onsuming steps are then learly the generalized Shur algorithm and the
produt (6).
2
4
The generalized Shur algorithm for rank{deient matries
Our desription of the generalized Shur algorithm allows to handle rank deient matries H T H: In this ase we an drop some rows: of the generator
during the algorithm. For this we need a tolerane, say Æ = kH T H k where
is the requested relative auray. Referring to the desription of setion 2,
we test if x y Æ: We then hek as well if the leading row a of the
urrent Shur omplement is small. If so, the urrently omputed row of the
Cholesky fator is negletable and we delete the two orresponding rows of the
generator. It is possible that a is muh larger than Æ although a Æ: In
this ase the deletion of a row of the Cholesky fator will yield residual errors
kH T H RT Rk of the same size. This is analyzed in this setion. From the rst
example we an onlude that the desribed proedure works aurately when
it is applied to a matrix H with a suÆiently large gap between signiant
singular values and negligible ones. On the other hand, a loss of auray in
the omputed fator R is observed when the distribution of the small singular
values of H shows a uniform and slow derease. The relative auray is
hosen equal to 10 in both examples.
Example 1 Consider the matrix H = [U T jY T ℄; with Y = U , where the rst
2
11
2
11
12
12
11
13
row and the last olumn of U are
[ 40 39 38 : : : 3 2 1 2 2 3 ℄;
[ 3 2 2 1 2 3 4 5 6 7 ℄T ;
respetively. The rank of the matrix H is 5 and kH T H k = 6:31 10 :
1
5
In Table 1 the results of the omputation of the R fator of the matrix H by
means of the standard QR and the generalized Shur algorithm are shown. We
denote by RM ; RS ; bakward error R ; and numerial rank, the R fator of
10
5
10
0
10
−5
10
−10
10
−15
10
−20
10
−25
10
−30
10
−35
10
0
2
4
6
8
10
12
14
16
18
20
Fig. 1. Distribution of the singular values, in logarithmi sale, of the matrix on-
sidered in Example 1
Table 2
Numerial results for Example 2
R
M
# ops
12:49 106
R
S
bakward error RM
bakward error RS
numerial rank
# ops
40:61 104
1:27 10
2:92 10
14
2
18
the QR fatorization of H omputed by the matlab funtion triu(qr(H)) and
by the generalized Shur algorithm, the bakward error of H T H dened as
kH T H RT Rk ;
kH T H k
1
1
and the rank of H deteted by the generalized Shur algorithm, respetively.
In this ase, the R fator is aurately omputed by the generalized Shur
algorithm, beause of the big dierene between the signiant singular values
and the negligible ones of H (see Figure 1).
Example 2 This is the fourth appliation onsidered in the next setion. In
Figure 2 we an see that the distribution of the small singular values of the
involved matrix H slightly dereases. We point out that the orrelation matrix
H T H omputed by matlab is not numerially s.p.d. beause of the nearly rank
deieny of H: Furthermore kH T H k1 = 3:99 104 : So, in this ase the
fast Cholesky fatorization, exploiting the blok{Hankel struture of H and
desribed in [12℄, an not be used. In Table 2 we an see that, although the
generalized Shur algorithm is very fast w.r.t. the standard QR algorithm, the
ahieved auray is not satisfatory.
11
3
10
2
10
1
10
0
10
−1
10
−2
10
−3
10
−4
10
−5
10
−6
10
−7
10
0
10
20
30
40
50
60
70
80
Fig. 2. Distribution of the singular values, in logarithmi sale, of the matrix on-
sidered in Example 2
5 Numerial results
In this setion results omputing the R matrix by means of the generalized
Shur algorithm are summarized. The data sets onsidered are publily available on the DAISY web site
http://www.esat.kuleuven.a.be/sista/daisy
At eah iteration of the generalized Shur algorithm, two Householder matries
and one modied hyperboli rotation are omputed in order to redue the
generator in proper form. All the numerial results have been obtained on a
Sun workstation Ultra 5 using Matlab 5.3.
Table 3 gives a summary desription of the appliations onsidered in our
omparison, indiating the number of inputs m; the number of outputs l; the
number of blok rows s; the total number of data samples used t and the
number of rows of H .
In Table 4 some results for the omputation of the R fator of the QR fatorization of H are presented. Rel. residual denotes
kjRM j jRS jk :
kjRM jk
1
1
The results in Table 4 are omparable with those desribed in [12℄, where the
R fator is obtained onsidering the Cholesky fatorization of the orrelation
matrix H T H , and exploiting the blok-Hankel struture of H . The analysis of
the fourth appliation is desribed in Example 2 of the previous setion.
12
Table 3
Summary desription of appliations.
Appl. # Appliation
m
l
s
t
N
1
Glass tubes
2
2 20 1401 1361
2
Labo dryer
1
1 15 1000
3
Glass oven
3
6 10 1247 1227
4
Mehanial utter
1
1 20 1024
960
5
Flexible robot arm
1
1 20 1024
984
6
Evaporator
3
3 10 6305 6285
7
CD player arm
2
2 15 2048 2018
8
Ball and beam
1
1 20 1000
9
Wall temperature
2
1 20 6800 1640
Table 4
Comparative results for the omputation of the
Appl. # Appliation
1
Glass tubes
2
Labo dryer
3
Glass oven
4
Mehanial utter
R
M
R
R
S
# ops
6:76 107
2:61 106
7:63 107
1:25 107
5
Flexible robot arm 1:25 107
6
Evaporator
7
CD player arm
8
Ball and beam
9
Wall temperature
1:82 108
5:77 107
1:22 107
4:67 107
960
fator.
# ops
7:01 106
970
3:36 105
6:38 106
4:89 103
4:76 105
1:11 107
2:59 106
4:67 105
1:64 106
bakward error RS
2:20 10
8:30 10
3:73 10
2:92 10
4:13 10
6:26 10
5:01 10
7:59 10
2:45 10
15
15
15
2
15
15
15
15
14
Rel.residual
8:73 10
9:48 10
7:91 10
13
12
0:39 100
3:38 10
5:13 10
2:08 10
6:79 10
3:76 10
6 Conlusions
In this paper the generalized Shur algorithm to ompute the R fator of the
QR fatorization of blok{Hankel matries, arising in some subspae identiation problems, is desribed.
It is shown that the generalized Shur algorithm is signiantly faster than
the lasssial QR fatorization. A rank{revealing implementation of the generalized Shur algorithm in ase of rank{deient matries is also disussed.
13
14
5
14
8
13
12
Algorithmi details and numerial results have been presented.
Referenes
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