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<HTML>
<HEAD><TITLE>IB01PX - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="IB01PX">IB01PX</A></H2>
<H3>
Estimating system matrices B and D using Kronecker products
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>
<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
To build and solve the least squares problem T*X = Kv, and
estimate the matrices B and D of a linear time-invariant (LTI)
state space model, using the solution X, and the singular
value decomposition information and other intermediate results,
provided by other routines.
The matrix T is computed as a sum of Kronecker products,
T = T + kron(Uf(:,(i-1)*m+1:i*m),N_i), for i = 1 : s,
(with T initialized by zero), where Uf is the triangular
factor of the QR factorization of the future input part (see
SLICOT Library routine IB01ND), N_i is given by the i-th block
row of the matrix
[ Q_11 Q_12 ... Q_1,s-2 Q_1,s-1 Q_1s ] [ I_L 0 ]
[ Q_12 Q_13 ... Q_1,s-1 Q_1s 0 ] [ ]
N = [ Q_13 Q_14 ... Q_1s 0 0 ] * [ ],
[ : : : : : ] [ ]
[ Q_1s 0 ... 0 0 0 ] [ 0 GaL ]
and where
[ -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 GaL is built from the first n
relevant singular vectors, GaL = Un(1:L(s-1),1:n), computed
by IB01ND.
The vector Kv is vec(K), with the matrix K defined by
K = [ K_1 K_2 K_3 ... K_s ],
where K_i = K(:,(i-1)*m+1:i*m), i = 1:s, is (n+L)-by-m.
The given matrices are Uf, GaL, and
[ L_1|1 ... L_1|s ]
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, computed by
SLICOT Library routine IB01PD.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE IB01PX( JOB, NOBR, N, M, L, UF, LDUF, UN, LDUN, UL,
$ LDUL, PGAL, LDPGAL, K, LDK, R, LDR, X, B, LDB,
$ D, LDD, TOL, IWORK, DWORK, LDWORK, IWARN,
$ INFO )
C .. Scalar Arguments ..
DOUBLE PRECISION TOL
INTEGER INFO, IWARN, L, LDB, LDD, LDK, LDPGAL, LDR,
$ LDUF, LDUL, LDUN, LDWORK, M, N, NOBR
CHARACTER JOB
C .. Array Arguments ..
DOUBLE PRECISION B(LDB, *), D(LDD, *), DWORK(*), K(LDK, *),
$ PGAL(LDPGAL, *), R(LDR, *), UF(LDUF, *),
$ UL(LDUL, *), UN(LDUN, *), X(*)
INTEGER IWORK( * )
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
JOB CHARACTER*1
Specifies which of 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.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
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.
UF (input/output) DOUBLE PRECISION array, dimension
( LDUF,M*NOBR )
On entry, the leading M*NOBR-by-M*NOBR upper triangular
part of this array must contain the upper triangular
factor of the QR factorization of the future input part,
as computed by SLICOT Library routine IB01ND.
The strict lower triangle need not be set to zero.
On exit, the leading M*NOBR-by-M*NOBR upper triangular
part of this array is unchanged, and the strict lower
triangle is set to zero.
LDUF INTEGER
The leading dimension of the array UF.
LDUF >= MAX( 1, M*NOBR ).
UN (input) DOUBLE PRECISION array, dimension ( LDUN,N )
The leading L*(NOBR-1)-by-N part of this array must
contain the matrix GaL, i.e., the leading part of the
first N columns of the matrix Un of relevant singular
vectors.
LDUN INTEGER
The leading dimension of the array UN.
LDUN >= L*(NOBR-1).
UL (input/output) DOUBLE PRECISION array, dimension
( LDUL,L*NOBR )
On entry, the leading (N+L)-by-L*NOBR part of this array
must contain the given matrix L.
On exit, if M > 0, the leading (N+L)-by-L*NOBR part of
this array is overwritten by the matrix
[ Q_11 Q_12 ... Q_1,s-2 Q_1,s-1 Q_1s ].
LDUL INTEGER
The leading dimension of the array UL. LDUL >= N+L.
PGAL (input) DOUBLE PRECISION array, dimension
( LDPGAL,L*(NOBR-1) )
The leading N-by-L*(NOBR-1) part of this array must
contain the pseudoinverse of the matrix GaL, computed by
SLICOT Library routine IB01PD.
LDPGAL INTEGER
The leading dimension of the array PGAL. LDPGAL >= N.
K (input) DOUBLE PRECISION array, dimension ( LDK,M*NOBR )
The leading (N+L)-by-M*NOBR part of this array must
contain the given matrix K.
LDK INTEGER
The leading dimension of the array K. LDK >= N+L.
R (output) DOUBLE PRECISION array, dimension ( LDR,M*(N+L) )
The leading (N+L)*M*NOBR-by-M*(N+L) part of this array
contains details of the complete orthogonal factorization
of the coefficient matrix T of the least squares problem
which is solved for getting the system matrices B and D.
LDR INTEGER
The leading dimension of the array R.
LDR >= MAX( 1, (N+L)*M*NOBR ).
X (output) DOUBLE PRECISION array, dimension
( (N+L)*M*NOBR )
The leading M*(N+L) elements of this array contain the
least squares solution of the system T*X = Kv.
The remaining elements are used as workspace (to store the
corresponding part of the vector Kv = vec(K)).
B (output) DOUBLE PRECISION array, dimension ( LDB,M )
The leading N-by-M part of this array contains the system
input matrix.
LDB INTEGER
The leading dimension of the array B. LDB >= N.
D (output) DOUBLE PRECISION array, dimension ( LDD,M )
If JOB = 'D', the leading L-by-M part of this array
contains the system input-output matrix.
If JOB = 'B', this array is not referenced.
LDD INTEGER
The leading dimension of the array D.
LDD >= L, if JOB = 'D';
LDD >= 1, if JOB = 'B'.
</PRE>
<B>Tolerances</B>
<PRE>
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).
</PRE>
<B>Workspace</B>
<PRE>
IWORK INTEGER array, dimension ( M*(N+L) )
DWORK DOUBLE PRECISION array, dimension ( LDWORK )
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK, and, if M > 0, DWORK(2) contains the
reciprocal condition number of the triangular factor of
the matrix T.
On exit, if INFO = -26, DWORK(1) returns the minimum
value of LDWORK.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( (N+L)*(N+L), 4*M*(N+L)+1 ).
For good performance, LDWORK should be larger.
</PRE>
<B>Warning Indicator</B>
<PRE>
IWARN INTEGER
= 0: no warning;
= 4: the least squares problem to be solved has a
rank-deficient coefficient matrix.
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
The matrix T is computed, evaluating the sum of Kronecker
products, and then the linear system T*X = Kv is solved in a
least squares sense. The matrices B and D are then directly
obtained from the least squares solution.
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[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.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE>
The implemented method is numerically stable.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
None
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
None
</PRE>
<B>Program Data</B>
<PRE>
None
</PRE>
<B>Program Results</B>
<PRE>
None
</PRE>
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