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 [4],
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 [4]. The matrices B and D are then obtained by solving certain linear systems in a least squares sense.References
[1] 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.
[2] 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.
[3] Van Overschee, P.
Subspace Identification : Theory - Implementation -
Applications.
Ph. D. Thesis, Department of Electrical Engineering,
Katholieke Universiteit Leuven, Belgium, Feb. 1995.
[4] 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.Further Comments
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
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