## IB01PY

### Estimating system matrices B and D using a structure exploiting QR factorization

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

```  1. To compute the triangular  (QR)  factor of the  p-by-L*s
structured matrix  Q,

[ Q_1s  Q_1,s-1  Q_1,s-2  ...  Q_12  Q_11 ]
[  0      Q_1s   Q_1,s-1  ...  Q_13  Q_12 ]
Q = [  0       0       Q_1s   ...  Q_14  Q_13 ],
[  :       :        :           :     :   ]
[  0       0        0     ...   0    Q_1s ]

and apply the transformations to the p-by-m matrix  Kexpand,

[ K_1 ]
[ K_2 ]
Kexpand = [ K_3 ],
[  :  ]
[ K_s ]

where, for MOESP approach (METH = 'M'), p = s*(L*s-n), and
Q_1i = u2(L*(i-1)+1:L*i,:)'  is  (Ls-n)-by-L,  for  i = 1:s,
u2 = Un(1:L*s,n+1:L*s),  K_i = K(:,(i-1)*m+1:i*m)  (i = 1:s)
is  (Ls-n)-by-m, and for N4SID approach (METH = 'N'), p = s*(n+L),
and

[   -L_1|1    ]          [ M_i-1 - L_1|i ]
Q_11 = [             ],  Q_1i = [               ],  i = 2:s,
[ I_L - L_2|1 ]          [     -L_2|i    ]

are  (n+L)-by-L  matrices, and
K_i = K(:,(i-1)*m+1:i*m),  i = 1:s,  is  (n+L)-by-m.
The given matrices are:
For  METH = 'M',  u2 = Un(1:L*s,n+1:L*s),
K(1:Ls-n,1:m*s);

[ L_1|1  ...  L_1|s ]
For  METH = 'N',  L = [                   ],   (n+L)-by-L*s,
[ L_2|1  ...  L_2|s ]

M = [ M_1  ...  M_s-1 ],  n-by-L*(s-1),  and
K,                        (n+L)-by-m*s.
Matrix M is the pseudoinverse of the matrix GaL,
built from the first  n  relevant singular
vectors,  GaL = Un(1:L(s-1),1:n),  and computed
by SLICOT Library routine IB01PD for METH = 'N'.

Matrix  Q  is triangularized  (in  R),  exploiting its structure,
and the transformations are applied from the left to  Kexpand.

2. To estimate the matrices B and D of a linear time-invariant
(LTI) state space model, using the factor  R,  transformed matrix
Kexpand, and the singular value decomposition information provided
by other routines.

IB01PY  routine is intended for speed and efficient use of the
memory space. It is generally not recommended for  METH = 'N',  as
IB01PX  routine can produce more accurate results.

```
Specification
```      SUBROUTINE IB01PY( METH, JOB, NOBR, N, M, L, RANKR1, UL, LDUL,
\$                   R1, LDR1, TAU1, PGAL, LDPGAL, K, LDK, R, LDR,
\$                   H, LDH, B, LDB, D, LDD, TOL, IWORK, DWORK,
\$                   LDWORK, IWARN, INFO )
C     .. Scalar Arguments ..
DOUBLE PRECISION   TOL
INTEGER            INFO, IWARN, L, LDB, LDD, LDH, LDK, LDPGAL,
\$                   LDR, LDR1, LDUL, LDWORK, M, N, NOBR, RANKR1
CHARACTER          JOB, METH
C     .. Array Arguments ..
DOUBLE PRECISION   B(LDB, *), D(LDD, *), DWORK(*), H(LDH, *),
\$                   K(LDK, *), PGAL(LDPGAL, *), R(LDR, *),
\$                   R1(LDR1, *), TAU1(*), UL(LDUL, *)
INTEGER            IWORK( * )

```
Arguments

Mode Parameters

```  METH    CHARACTER*1
Specifies the subspace identification method to be used,
as follows:
= 'M':  MOESP  algorithm with past inputs and outputs;
= 'N':  N4SID  algorithm.

JOB     CHARACTER*1
Specifies whether or not the matrices B and D should be
computed, as follows:
= 'B':  compute the matrix B, but not the matrix D;
= 'D':  compute both matrices B and D;
= 'N':  do not compute the matrices B and D, but only the
R  factor of  Q  and the transformed Kexpand.

```
Input/Output Parameters
```  NOBR    (input) INTEGER
The number of block rows,  s,  in the input and output
Hankel matrices processed by other routines.  NOBR > 1.

N       (input) INTEGER
The order of the system.  NOBR > N > 0.

M       (input) INTEGER
The number of system inputs.  M >= 0.

L       (input) INTEGER
The number of system outputs.  L > 0.

RANKR1  (input) INTEGER
The effective rank of the upper triangular matrix  r1,
i.e., the triangular QR factor of the matrix  GaL,
computed by SLICOT Library routine IB01PD. It is also
the effective rank of the matrix  GaL.  0 <= RANKR1 <= N.
If  JOB = 'N',  or  M = 0,  or  METH = 'N',  this
parameter is not used.

UL      (input/workspace) DOUBLE PRECISION array, dimension
( LDUL,L*NOBR )
On entry, if  METH = 'M',  the leading  L*NOBR-by-L*NOBR
part of this array must contain the matrix  Un  of
relevant singular vectors. The first  N  columns of  UN
need not be specified for this routine.
On entry, if  METH = 'N',  the leading  (N+L)-by-L*NOBR
part of this array must contain the given matrix  L.
On exit, the leading  LDF-by-L*(NOBR-1) part of this array
is overwritten by the matrix  F  of the algorithm in ,
where  LDF = MAX( 1, L*NOBR-N-L ), if  METH = 'M';
LDF = N,                    if  METH = 'N'.

LDUL    INTEGER
The leading dimension of the array  UL.
LDUL >= L*NOBR, if  METH = 'M';
LDUL >= N+L,    if  METH = 'N'.

R1      (input) DOUBLE PRECISION array, dimension ( LDR1,N )
If  JOB <> 'N',  M > 0,  METH = 'M',  and  RANKR1 = N,
the leading  L*(NOBR-1)-by-N  part of this array must
contain details of the QR factorization of the matrix
GaL,  as computed by SLICOT Library routine IB01PD.
Specifically, the leading N-by-N upper triangular part
must contain the upper triangular factor  r1  of  GaL,
and the lower  L*(NOBR-1)-by-N  trapezoidal part, together
with array TAU1, must contain the factored form of the
orthogonal matrix  Q1  in the QR factorization of  GaL.
If  JOB = 'N',  or  M = 0,  or  METH = 'N', or  METH = 'M'
and  RANKR1 < N,  this array is not referenced.

LDR1    INTEGER
The leading dimension of the array  R1.
LDR1 >= L*(NOBR-1), if  JOB <> 'N',  M > 0,  METH = 'M',
and  RANKR1 = N;
LDR1 >= 1,          otherwise.

TAU1    (input) DOUBLE PRECISION array, dimension ( N )
If  JOB <> 'N',  M > 0,  METH = 'M',  and  RANKR1 = N,
this array must contain the scalar factors of the
elementary reflectors used in the QR factorization of the
matrix  GaL,  computed by SLICOT Library routine IB01PD.
If  JOB = 'N',  or  M = 0,  or  METH = 'N', or  METH = 'M'
and  RANKR1 < N,  this array is not referenced.

PGAL    (input) DOUBLE PRECISION array, dimension
( LDPGAL,L*(NOBR-1) )
If  METH = 'N',  or  JOB <> 'N',  M > 0,  METH = 'M'  and
RANKR1 < N,  the leading  N-by-L*(NOBR-1)  part of this
array must contain the pseudoinverse of the matrix  GaL,
as computed by SLICOT Library routine IB01PD.
If  METH = 'M'  and  JOB = 'N',  or  M = 0,  or
RANKR1 = N,  this array is not referenced.

LDPGAL  INTEGER
The leading dimension of the array  PGAL.
LDPGAL >= N,  if   METH = 'N',  or  JOB <> 'N',  M > 0,
and  METH = 'M'  and RANKR1 < N;
LDPGAL >= 1,  otherwise.

K       (input/output) DOUBLE PRECISION array, dimension
( LDK,M*NOBR )
On entry, the leading  (p/s)-by-M*NOBR  part of this array
must contain the given matrix  K  defined above.
On exit, the leading  (p/s)-by-M*NOBR  part of this array
contains the transformed matrix  K.

LDK     INTEGER
The leading dimension of the array  K.  LDK >= p/s.

R       (output) DOUBLE PRECISION array, dimension ( LDR,L*NOBR )
If  JOB = 'N',  or  M = 0,  or  Q  has full rank, the
leading  L*NOBR-by-L*NOBR  upper triangular part of this
array contains the  R  factor of the QR factorization of
the matrix  Q.
If  JOB <> 'N',  M > 0,  and  Q  has not a full rank, the
leading  L*NOBR-by-L*NOBR  upper trapezoidal part of this
array contains details of the complete orhogonal
factorization of the matrix  Q,  as constructed by SLICOT
Library routines MB03OD and MB02QY.

LDR     INTEGER
The leading dimension of the array  R.  LDR >= L*NOBR.

H       (output) DOUBLE PRECISION array, dimension ( LDH,M )
If  JOB = 'N'  or  M = 0,  the leading  L*NOBR-by-M  part
of this array contains the updated part of the matrix
Kexpand  corresponding to the upper triangular factor  R
in the QR factorization of the matrix  Q.
If  JOB <> 'N',  M > 0,  and  METH = 'N'  or  METH = 'M'
and  RANKR1 < N,  the leading  L*NOBR-by-M  part of this
array contains the minimum norm least squares solution of
the linear system  Q*X = Kexpand,  from which the matrices
B  and  D  are found. The first  NOBR-1  row blocks of  X
appear in the reverse order in  H.
If  JOB <> 'N',  M > 0,  METH = 'M'  and  RANKR1 = N,  the
leading  L*(NOBR-1)-by-M  part of this array contains the
matrix product  Q1'*X,  and the subarray
L*(NOBR-1)+1:L*NOBR-by-M  contains the  corresponding
submatrix of  X,  with  X  defined in the phrase above.

LDH     INTEGER
The leading dimension of the array  H.  LDH >= L*NOBR.

B       (output) DOUBLE PRECISION array, dimension ( LDB,M )
If  M > 0,  JOB = 'B' or 'D'  and  INFO = 0,  the leading
N-by-M part of this array contains the system input
matrix.
If  M = 0  or  JOB = 'N',  this array is not referenced.

LDB     INTEGER
The leading dimension of the array B.
LDB >= N, if  M > 0 and JOB = 'B' or 'D';
LDB >= 1, if  M = 0 or  JOB = 'N'.

D       (output) DOUBLE PRECISION array, dimension ( LDD,M )
If  M > 0,  JOB = 'D'  and  INFO = 0,  the leading
L-by-M part of this array contains the system input-output
matrix.
If  M = 0  or  JOB = 'B'  or  'N',  this array is not
referenced.

LDD     INTEGER
The leading dimension of the array D.
LDD >= L, if  M > 0 and JOB = 'D';
LDD >= 1, if  M = 0 or  JOB = 'B' or 'N'.

```
Tolerances
```  TOL     DOUBLE PRECISION
The tolerance to be used for estimating the rank of
matrices. If the user sets  TOL > 0,  then the given value
of  TOL  is used as a lower bound for the reciprocal
condition number;  an m-by-n matrix whose estimated
condition number is less than  1/TOL  is considered to
be of full rank.  If the user sets  TOL <= 0,  then an
implicitly computed, default tolerance, defined by
TOLDEF = m*n*EPS,  is used instead, where  EPS  is the
relative machine precision (see LAPACK Library routine
DLAMCH).
This parameter is not used if  M = 0  or  JOB = 'N'.

```
Workspace
```  IWORK   INTEGER array, dimension ( LIWORK )
where  LIWORK >= 0,       if  JOB =  'N',  or   M = 0;
LIWORK >= L*NOBR,  if  JOB <> 'N',  and  M > 0.

DWORK   DOUBLE PRECISION array, dimension ( LDWORK )
On exit, if  INFO = 0,  DWORK(1) returns the optimal value
of  LDWORK,  and, if  JOB <> 'N',  and  M > 0,  DWORK(2)
contains the reciprocal condition number of the triangular
factor of the matrix  R.
On exit, if  INFO = -28,  DWORK(1)  returns the minimum
value of LDWORK.

LDWORK  INTEGER
The length of the array DWORK.
LDWORK >= MAX( 2*L, L*NOBR, L+M*NOBR ),
if  JOB = 'N',  or  M = 0;
LDWORK >= MAX( L+M*NOBR, L*NOBR + MAX( 3*L*NOBR+1, M ) ),
if  JOB <> 'N',  and  M > 0.
For good performance,  LDWORK  should be larger.

```
Warning Indicator
```  IWARN   INTEGER
= 0:  no warning;
= 4:  the least squares problem to be solved has a
rank-deficient coefficient matrix.

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value;
= 3:  a singular upper triangular matrix was found.

```
Method
```  The QR factorization is computed exploiting the structure,
as described in .
The matrices  B  and  D  are then obtained by solving certain
linear systems in a least squares sense.

```
References
```   Verhaegen M., and Dewilde, P.
Subspace Model Identification. Part 1: The output-error
state-space model identification class of algorithms.
Int. J. Control, 56, pp. 1187-1210, 1992.

 Van Overschee, P., and De Moor, B.
N4SID: Two Subspace Algorithms for the Identification
of Combined Deterministic-Stochastic Systems.
Automatica, Vol.30, No.1, pp. 75-93, 1994.

 Van Overschee, P.
Subspace Identification : Theory - Implementation -
Applications.
Ph. D. Thesis, Department of Electrical Engineering,
Katholieke Universiteit Leuven, Belgium, Feb. 1995.

 Sima, V.
Subspace-based Algorithms for Multivariable System
Identification.
Studies in Informatics and Control, 5, pp. 335-344, 1996.

```
Numerical Aspects
```  The implemented method for computing the triangular factor and
updating Kexpand is numerically stable.

```
```  The computed matrices B and D are not the least squares solutions
delivered by either MOESP or N4SID algorithms, except for the
special case n = s - 1, L = 1. However, the computed B and D are
frequently good enough estimates, especially for  METH = 'M'.
Better estimates could be obtained by calling SLICOT Library
routine IB01PX, but it is less efficient, and requires much more
workspace.

```
Example

Program Text

```  None
```
Program Data
```  None
```
Program Results
```  None
```