Automatica 42 (2006) 741–746www.elsevier.com/locate/automatica

Brief Paper

A stochastic realization algorithm via block LQ decompositionin Hilbert space�

Hideyuki Tanaka∗, Tohru KatayamaDepartment of Applied Mathematics and Physics, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan

Received 3 March 2004; received in revised form 10 July 2005; accepted 6 December 2005Available online 6 March 2006

Abstract

A stochastic realization problem of a stationary stochastic process is re-visited, and a new stochastically balanced realization algorithm isderived in a Hilbert space generated by second-order stationary processes. The present algorithm computes a stochastically balanced realizationby means of the singular value decomposition of a weighted block Hankel matrix derived by a “block LQ decomposition”. Extension to astochastic subspace identification method explains how the proposed abstract algorithm is implemented in system identification.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Stochastic realization; Subspace identification; Canonical correlation analysis; Innovation representation; Hilbert space; LQ decomposition

1. Introduction

The stochastic realization problem is to find a set of Markovmodels whose output covariance matrix matches a given co-variance sequence of a stationary random process (Faurre,1976; Faurre, Clerget, & Germain, 1979). A novel method ofstochastic realization has been developed by using the canoni-cal correlation analysis (CCA) (Akaike, 1974, 1975), and thena stochastic model reduction technique has been derived (Desai& Pal, 1984; Desai, Pal, & Kirkpatrick, 1985) via the stochas-tically balanced realization. Also, an approximation algorithmof stochastic systems has been developed based on the inter-nally balanced realization (Arun & Kung, 1990). Moreover,state space identification methods for the time series have ex-tensively been studied based on the CCA (Lindquist & Picci,1996a), including non-trivial problems related to the positivityof covariance matrices. The conditional CCA is employed to

� Part of this paper was presented at the 13th IFAC Symposium on SystemIdentification (SYSID2003) which was held in Rotterdam, The Netherlandsduring August 2003. This paper was recommended for publication in revisedform by Associate Editor Hitay Ozbay under the direction of Editor IanPetersen.

∗ Corresponding author. Tel.: +81 75 753 4754; fax: +81 753 5507.E-mail addresses: htanaka@amp.i.kyoto-u.ac.jp (H. Tanaka),

katayama@amp.i.kyoto-u.ac.jp (T. Katayama).

0005-1098/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2005.12.025

derive a stochastic realization algorithm in the presence ofexogenous inputs (Katayama & Picci, 1999).

It is well-known (Gevers, 2003; Katayama, 2005) thatstochastic realization theory has played a basic role in derivingstochastic subspace identification algorithms (Van Overschee& De Moor, 1993, 1996), which identify stochastic state spacemodels from finite strings of time-series data. But, subspaceidentification methods have brought a nontrivial problemrelated to positivity, since only a finite inexact covariancesequence is available from finite time-series data; therebythe associated Riccati equation may not have the stabilizingsolution. From these observations, state space identificationmethods (Aoki, 1990; Van Overschee & De Moor, 1993) maynot work for arbitrary data. Lindquist and Picci (1996a, 1996b)have thus analyzed state space identification algorithms in thelight of the geometric theory of stochastic realization. Theyhave discussed the state space modeling of time series basedon the three different assumptions that (i) an infinite exactcovariance sequence is available, (ii) a finite exact covariancesequence is available, and (iii) a finite string of time-series datais available.

The objective of this paper is to avoid solving Riccatiequations in a stochastic realization algorithm; we establish afoundation for deriving stochastic subspace identification algo-rithms adopting a procedure different from the Faurre’s (1976)realization algorithm. To this end, we develop a realization

742 H. Tanaka, T. Katayama / Automatica 42 (2006) 741–746

algorithm in a Hilbert space of a second-order stationary timeseries, rather than the classical approach based on covariancematrices (Desai & Pal, 1984). We thus assume that an infinitestring of time series is given, or equivalently that (i) an infinitecovariance sequence is available. We hence obtain a block LQdecomposition in the Hilbert space and present a new algorithmto find the forward innovation representation. We furthermoreadapt the present realization algorithm to a stochastic subspaceidentification method based on the assumption that (iii) a finitestring of data is given.

The paper is organized as follows. Section 2 reviews thestochastic realization theory, and Section 3 introduces a Hilbertspace. Section 4 gives a block LQ decomposition, and Sec-tion 5 derives a new stochastic realization algorithm. Section6 discusses its extension to a stochastic subspace identificationmethod. Conclusions are stated in Section 7. Appendix A in-cludes proofs of Lemmas and Theorems.

2. Preliminaries

Consider a second-order stationary process {yt , t =0, ±1, ±2, . . .}, where yt is a p-dimensional non-deterministicprocess with mean zero and covariance matrices

�k = E{yt+ky�t }, k = 0, ±1, ±2, . . . . (1)

We assume that �k (k = 0, ±1, . . .) is a positive real sequence,∑i,j u�

i �i−j uj > 0 (ui /≡ 0), or more precisely, for every t =1, 2, 3, . . ., the block Toeplitz matrix,

�t :=

⎡⎢⎢⎣

�0 �1 · · · �t−1��

1 �0 · · · �t−2...

.... . .

...

��t−1 ��

t−2 · · · �0

⎤⎥⎥⎦ ,

is positive definite (Lindquist & Picci, 1996a). We further as-sume that {�t }t �1 is coercive, i.e. �t > �I for some � > 0 andall t �1.

The spectral density of yt can be computed as an ordinaryFourier transform Υ (z)=∑∞

j=−∞ �j z−j . We may write Υ (z)=

�(z) + ��(z−1) where �(z) is the causal component of Υ (z)

given by �(z)= 12�0 +�1z

−1 +�2z−2 +· · ·. The positivity and

coercivity conditions for a covariance sequence is equivalentto positive semi-definiteness and definiteness of Υ (z) on theunit circle (Lindquist & Picci, (1996a,b)), i.e. Υ (ei�)�0 andΥ (ei�) > 0 for � ∈ [−�, �], respectively.

Suppose that there exists a finite dimensional realization foryt , so that the covariance matrix has a decomposition �k =HFk−1� (k = 1, 2, . . .), where we assume that (F , �, H ) isa minimal realization with F ∈ Rn×n stable. In terms of (F ,�, H ), �(z) can be expressed as (Lindquist & Picci, 1996b)�(z) = H(zI − F)−1� + 1

2�0.We introduce the stochastically balanced realization via

the CCA along the line of the classical algorithm (Desai &Pal, 1984) which is rather based on covariance matrices. De-fine stacked vectors as Y−(t) = [y�

t−1 y�t−2 y�

t−3 · · · ]� and

Y+(t) = [y�t y�

t+1 y�t+2 · · · ]�, where Y−(t) and Y+(t) are,

respectively, called the past and the future. We define moreovermatrices as1

� :=

⎡⎢⎢⎢⎣

�0 �1 �2 · · ·��

1 �0 �1 · · ·��

2 ��1 �0 · · ·

......

.... . .

⎤⎥⎥⎥⎦ = E{Y−(t)Y−(t)�},

� :=

⎡⎢⎢⎣

�0 ��1 ��

2 · · ·�1 �0 ��

1 · · ·�2 �1 �0 · · ·...

......

. . .

⎤⎥⎥⎦ = E{Y+(t)Y+(t)�},

H :=

⎡⎢⎢⎣

�1 �2 �3 · · ·�2 �3 �4 · · ·�3 �4 �5 · · ·...

......

. . .

⎤⎥⎥⎦ = E{Y+(t)Y−(t)�}.

We compute the singular value decomposition (SVD) of thenormalized Hankel matrix H as2

�−1/2H�−�/2 = U V �, ∈ Rn×n. (2)

Define extended observability and reachability matrices, re-spectively, as

O := �1/2U 1/2, C := 1/2V ���/2. (3)

The Hankel matrix H has then a canonical decomposition H=OC, where O and C are described by certain matrices A ∈Rn×n, G ∈ Rn×p and C ∈ Rp×n as

O = [C� (CA)� (CA2)� · · · ]�, (4)

C = [G AG A2G A3G · · ·], (5)

and �k has a decomposition �k = CAk−1G (k = 1, 2, . . .). Itfollows from (3) that rank O = rank C = rank = n and

C�−1C� = = O��−1O (6)

hold. By means of the matrix inversion lemma, the matrix satisfies (Desai & Pal, 1984) the forward Riccati equation

= AA� + (G − AC�)

× (�0 − CC�)−1(G − AC�)�, (7)

and the backward Riccati equation

= A�A + (C − B�A)�

× (�0 − B�B)−1(C − B�A).

1 The infinite matrices � and � have their inverses, respectively, sincewe have assumed that {�t }t � 1 is coercive.

2 For an infinite matrix A = (aij ) with A = A�, aij ∈ R, we define the

quadratic form x�Ax = ∑i,j xiaij xj , where x = [x1 x2 · · · ]�, xi ∈ R. If

x�Ax > 0 for all x /≡ 0, then A is strictly positive real and is written asA > 0. A square root of A denotes a matrix X with A = XX�, X = X�,where we describe A = A1/2A�/2. If A is invertible, so is A1/2.

H. Tanaka, T. Katayama / Automatica 42 (2006) 741–746 743

The realization (A, G, C) in (4) and (5) is thus stochasticallybalanced. Define matrices K and R as

K := (G − AC�)(�0 − CC�)−1, (8)

R := �0 − CC�. (9)

Then, A − KC is stable, and R is positive definite which isderived from � > 0 (Faurre, 1976).

Define a stochastically balanced state xt and an innovationprocess vt as xt =C�−1Y−(t) and vt = yt −Cxt , respectively.It can then be shown (Desai et al., 1985) that yt has a stochas-tically balanced realization

[xt+1yt

]=

[A K

C I

] [xt

vt

], (10)

where xt is the state with the covariance matrix , and vt isthe forward innovation with the covariance matrix R.

Given an infinite sequence of data {yt , t = 0, ±1, ±2, . . .},our problem is to develop a stochastic realization algorithm ina Hilbert space to find the forward innovation representation(10) without solving Riccati equations explicitly.

3. Hilbert space of sample functions

In this section, we introduce a Hilbert space of observedinfinite strings of data (Lindquist & Picci, 1996a,b). Let the tailmatrix be defined by

yt := [yt yt+1 yt+2 · · ·] ∈ Rp×∞.

We then define a vector space as Y∞ := {∑ a�k yk | ak ∈

Rp, k=0, ±1, . . .} which is a linear space spanned by all finitelinear combinations of row vectors of yt . For a�yi , b

�yj ∈ Y∞,define a bilinear form 〈·, ·〉 (inner product) as 〈a�yi , b

�yj 〉 :=a��i−j b. By completing the vector space Y∞ with the norminduced by the inner product 〈·, ·〉, we get a Hilbert space,which is also written as Y∞.

Let U be a Hilbert subspace of Y∞, and let the orthogonalprojection of y ∈ Y∞ onto the subspace U ⊆ Y∞ be denotedby E( y |U). The row space spanned by a matrix U is expressedas span(U) and the orthogonal projection is also written asE( y |U) := E( y | span (U)). We extend Y∞ to Y•×∞ so thatmatrices are included as its elements.

4. Block LQ decomposition for a forward realization

We define matrices as Y−t := [ y�

t−1 y�t−2 y�

t−3 · · · ]� andY+

t := [ y�t y�

t+1 y�t+2 · · · ]�. Defining also variables as

yt := E( yt |Y−t ), vt := yt − E(yt |Y−

t ), (11)

we have yt = yt + vt and 〈yt , vt 〉 = 0.

Lemma 1. The process vt ∈ Yp×∞ defined by (11) is an in-novation process satisfying

〈 yi , vj 〉 ={

0 (i < j),

Li−j R (i�j),(12)

〈vi , vj 〉 = Rij , (13)

where the matrices Lk are defined as

L0 = I, Lk = CAk−1K, k = 1, 2, . . . , (14)

and where (A, K, C) is computed from (4) and (8).

Proof. See Appendix A.1.

We derive a block LQ decomposition in the Hilbert space.In terms of Lk in (14), we define a block Hankel matrix as

S :=

⎡⎢⎢⎣

L1 L2 L3 · · ·L2 L3 L4 · · ·L3 L4 L5 · · ·...

......

. . .

⎤⎥⎥⎦ , (15)

and block upper and lower triangular matrices as

L− :=

⎡⎢⎢⎣

L0 L1 L2 · · ·L0 L1 · · ·

L0 · · ·0

. . .

⎤⎥⎥⎦ ,

L+ :=

⎡⎢⎢⎣

L0 0L1 L0L2 L1 L0...

......

. . .

⎤⎥⎥⎦ .

We also define matrices as V −t := [v�

t−1 v�t−2 v�

t−3 · · · ]� and

V +t := [v�

t v�t+1 v�

t+2 · · · ]�, and a block diagonal matrix as

R := block-diag(R, R, . . .).

Theorem 1. In terms of the innovation processes V −t and V +

t ,the past Y−

t and the future Y+t are decomposed as[

Y−t

Y+t

]=

[L− 0S L+

] [V −

t

V +t

], (16)

where V −t and V +

t satisfy⟨[V −

t

V +t

],

[V −

t

V +t

]⟩=

[R 00 R

]. (17)

Proof. See Appendix A.2.

The decomposition (16) can be equivalently expressed by ablock LQ decomposition as⎡⎢⎢⎢⎢⎢⎣

...

yt−1yt

yt+1...

⎤⎥⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎣

. . .

· · · L0· · · L1 L0· · · L2 L1 L0

......

.... . .

⎤⎥⎥⎥⎥⎥⎦

⎡⎢⎢⎢⎢⎢⎣

...

vt−1vt

vt+1...

⎤⎥⎥⎥⎥⎥⎦

. (18)

744 H. Tanaka, T. Katayama / Automatica 42 (2006) 741–746

It should be noted that Arun and Kung (1990) have deriveda method of computing an internally balanced realization fora scalar stationary process. In fact, they have derived Y+

t =SV −

t +L+V +t based on the forward innovation representation

(10). They have not however given an explicit description ofthe block LQ decomposition as in (16) or (18). One of essentialdifferences between their and our results is that we give theblock LQ decomposition (18) in terms of the innovation processvt which naturally leads to a stochastic subspace identificationalgorithm as shown in Section 6.

Corollary 1. The orthogonal projection of the future onto thepast is written as

E(Y+t |Y−

t ) = SV −t . (19)

Proof. From (11), we have span (Y−t )= span (V −

t ), and henceE(Y+

t |Y−t ) = E(Y+

t | V −t ). We therefore have (19) from (16)

and (17). �

By definition of the orthogonal projection, we have

E(Y+t |Y−

t ) = H�−1Y−t = Oxt , (20)

where xt is defined as

xt := C�−1Y−t . (21)

The orthogonal projection (20), originally derived by Akaike(1975) and Faurre (1976), is expressed by means of the statevector xt ∈ Yn×∞. A difference between the classical andpresent expressions is that the orthogonal projection (19) isdescribed in terms of the innovation process V −

t .

5. Stochastic realization algorithm

Observing that S in (15) is a block Hankel matrix formedby Lj in (14), we derive a stochastic realization algorithm. Wedefine a matrix as F := [K AK A2K · · ·] in terms of A andK in (14).

Theorem 2. The block Hankel matrix S has a decomposition

S = OF, (22)

where O is given by (4). The matrix S has thus rank n, andSRS� = H�−1H� holds.

Proof. See Appendix A.3.

We have �−1/2SRS��−�/2 = U 2U� from (2) and The-orem 2, and hence the following corollary.

Corollary 2. Given S, R and �, the SVD of weighted S canbe written as

�−1/2SR1/2 = U V �, ∈ Rn×n, (23)

where V �V = In. Thus, the matrices O and F are given by

O = �1/2U 1/2, F = 1/2V �R−1/2. (24)

Note that O derived above equals the matrix O in (3).

Based on the above decomposition results, we present astochastic balanced realization algorithm in the Hilbert spacewithout solving Riccati equations explicitly.

Stochastic realization algorithm.Step 1. Given Y−

t and Y+t , we compute V −

t , R and S as(16) and (17).

Step 2. Compute the SVD of the weighted S in (23), andobtain O and F from (24).

Step 3. Compute A from O(1 : ∞, :)A=O(p +1 : ∞, :) andset C = O(1 : p, :).

Step 4. Determine K and R as K=F(:, 1 : p) and R=R(1 :p, 1 : p), respectively.

The system (10) with the quadruple (A, K, C, R) obtainedabove is a forward innovation representation of yt .

6. Extension to a subspace identification method

Assume that a finite string of data {y0, y1, . . . , y2�+�−2} isavailable, with � and � large. Define matrices as

yt := [yt yt+1 · · · yt+�−1] ∈ Rp×�

for t =0, . . ., 2�−1, and Y+0 =[y�

0 y�1 · · · y�

2�−1]� ∈ R2�p×�.

Assuming that (Y+0 )(Y+

0 )� > 0 holds, we obtain a prototype ofa stochastic subspace identification algorithm.

A prototype stochastic subspace identification algorithm.Step 1: Given Y+

0 , we compute the decomposition3

Y+0 =

⎡⎣

L0,0 0...

. . .

L2�−1,0 · · · L2�−1,2�−1

⎤⎦

⎡⎣

v0...

v2�−1

⎤⎦ = L+

0 V +0 ,

(25)

with Li,j ∈ Rp×p and Li,i = Ip, where the matrix V +0 ∈

R2�p� satisfies (1/�)(V +0 )(V +

0 )� =R+0 , and R+

0 =block-diag(R0, . . . , R2�−1). Partition L+

0 ∈ R2�p×2�p as

L+0 =

[L− 0S L+

]

where L−, L+, S ∈ R�p×�p, and define matrices as Y+� :=

[ y�� y�

�+1 · · · y�2�−1]� ∈ R�p×� and � := 1/�(Y+

� )(Y+� )�.

3 The decomposition (25) is computed by means of the standard LQdecomposition. Moreover, Li,j → Li−j and R → R hold when � → ∞ and� → ∞ from Theorem 1.

H. Tanaka, T. Katayama / Automatica 42 (2006) 741–746 745

Step 2: Compute the SVD of the weighted S,

�−1/2SR1/2 = U V �, ∈ Rn×n,

where R = block-diag (R0, . . . , R�−1).Also define matrices as O = �1/2U 1/2 and F =

1/2V �R−1/2.Step 3: Compute A = O(1 : p(�−1), : )†O(p + 1 : p�, :),

where † expresses the Moore–Penrose generalized inverse. Alsoset C = O(1 : p, :).

Step 4: Define matrices as K = F(:, p(� − 1) + 1 : p�) andR = R+

0 (p(� − 1) + 1 : p�, p(� − 1) + 1 : p�).

As we have seen in Section 5, we have (A, K, C, R) →(A, K, C, R) when � → ∞ and � → ∞. The prototypestochastic subspace identification algorithm never stops com-puting, so that it always produces Markov models. This fact isone of the advantages of our algorithm, though it is a disad-vantage in that there is no indication for potential users that theresults may be different from what is expected, and that A andA − KC may be unstable.

The subspace identification method (Van Overschee & DeMoor, 1993), which is based on the Faurre’s (1976) stochasticrealization algorithm, solves a Riccati equation

P = AP A� + (G − AP C�)

× (�0 − CP C�)−1(G − AP C�)�, (26)

where (A, G, C) is given by covariance matrices �k ≈CAk−1G with �k ≈ �k := (1/�)yi+k y�

i . In this case, thereis a possibility that the Riccati equation (26) does not have astabilizing solution, since �(z) = C(zI − A)−1G + 1

2 �0 is notalways positive real due to inexact covariance matrices.

7. Conclusions

We have developed a new stochastic realization algorithmwithout solving Riccati equations explicitly, based on a blockLQ decomposition in the Hilbert space of observed infinitestrings of time series data. We have, moreover, discussed howthe proposed algorithm can be adapted to a finite string of time-series data.

Acknowledgment

This work was partially supported by the Ministry of Educa-tion, Culture, Sports, Science and Technology of Japan underGrant-in-Aid for Young Scientists (B), 15760316.

Appendix A. Proofs of Lemma and Theorems

We first provide two lemmas, which are easily proved.

Lemma 2. For i�j , 〈 yi , Y−j 〉 = CAi−jC holds.

Lemma 3. For yj and xj defined, respectively, as (11) and

(21), yj = CC�−1Y−j = Cxj holds.

A.1. Proof of Lemma 1

We prove (13) for i �= j . Since vj is orthogonal to span(Y−j )

from (11), we have 〈 Y−j , vj 〉 = 0, and hence

〈 yi , vj 〉 = 0 (i < j) (A.1)

and 〈 Y−i , vj 〉=0 for i < j . From Lemma 3, we moreover have

〈 yi , vj 〉 = CC�−1〈 Y−i , vj 〉 = 0 for i < j . It follows from (11)

that 〈 vi , vj 〉=〈 yi − yi , vj 〉=0 holds for i < j . We hence derive

〈 vi , vj 〉 = 0 (i �= j), (A.2)

since 〈 vi , vj 〉=〈 vj , vi〉� =0 holds for i > j . We prove (13) fori = j . We have 〈 xj , xj 〉=C�−1C� = from (21) and (6), andalso 〈 yj , yj 〉 = 〈 yj , yj 〉 + 〈 vj , vj 〉 from (11). We hence obtainfrom Lemma 3

〈 vj , vj 〉 = 〈 yj , yj 〉 − 〈 yj , yj 〉= �0 − C〈 xj , xj 〉C� = R, (A.3)

where we have used (9). From (A.2) and (A.3), we have (13).We next prove (12). Assume that i > j . By using (6),

Lemmas 2 and 3, we obtain 〈 yi , yj 〉 = 〈 yi , CC�−1Y−j 〉 =

CAi−jC�−1C�C� = CAi−j C�. It follows from �i−j =〈 yi , yj 〉 = CAi−j−1G that

〈 yi , vj 〉 = 〈 yi , yj 〉 − 〈 yi , yj 〉= CAi−j−1(G − AC�) = Li−j R, (A.4)

where (8), (9) and the definition Lk in (14) are used for (A.4).Let us consider (12) for i = j . From (11) and (A.3), we have〈 yi , vi〉=〈 yi+vi , vi〉=〈 vi , vi〉=R. From this equation togetherwith (A.1) and (A.4), we have (12), and complete the proof ofLemma 1. �

A.2. Proof of Theorem 1

From (13), we can easily derive (17). Using (12), we obtain⟨[Y−

t

Y+t

],

[V −

t

V +t

]⟩=

[L−R 0SR L+R

].

We therefore obtain (16), since we have from (11)[Y−

t

Y+t

]= E

([Y−

t

Y+t

]∣∣∣∣[Y−

t

Y+t

])= E

([Y−

t

Y+t

]∣∣∣∣[V −

t

V +t

]).

We have thus proved Theorem 1. �

A.3. Proof of Theorem 2

Consider a Lyapunov equation =AA� +KRK�, whichis obtained by substituting (8) and (9) into (7). The solutionto the Lyapunov equation is given by = FRF�, wherewe see that F has rank n since has full rank. We thushave (22) from (14) and (15), and also rank S = n, fromrank O= n and rank F= n. It follows from (19) and (20) that〈 SV −

t , SV −t 〉 = 〈H�−1Y−

t ,H�−1Y−t 〉 holds. We thus ob-

tain SRS� = H�−1H�. �

746 H. Tanaka, T. Katayama / Automatica 42 (2006) 741–746

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Hideyuki Tanaka received B.Sc. and M.Sc. de-grees in engineering from Kyoto University, in1993 and 1995, respectively, and Ph.D. degree(engineering) from Kyoto University, in 1999.Since 1998, he has been with Applied Mathe-matics and Physics, Kyoto University. His re-search interests include system identification androbust control.

Tohru Katayama received the B.E., M.E., andPh.D. degrees all in Applied Mathematics andPhysics from Kyoto University, in 1964, 1966,and 1969, respectively. He was a Professor atthe Department of Applied Mathematics andPhysics, Kyoto University from 1986 to 2005,and is now Professor at the Faculty of Cultureand Information Science, Doshisha University.He had visiting positions at UCLA from 1974 to1975, and at University of Padova in 1997. Hewas the Chair of IFAC Coordinating Commit-tee of Signals and Systems from 2002 to 2005,

and is now the Vice-Chair of IFAC Technical Board (2005–2008). His researchinterests include estimation theory, stochastic realization, subspace method ofidentification, blind identification, and control of industrial processes.