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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
C SLICOT RELEASE 5.0.
C
C Copyright (c) 2002-2009 NICONET e.V.
C
C This program is free software: you can redistribute it and/or
C modify it under the terms of the GNU General Public License as
C published by the Free Software Foundation, either version 2 of
C the License, or (at your option) any later version.
C
C This program is distributed in the hope that it will be useful,
C but WITHOUT ANY WARRANTY; without even the implied warranty of
C MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
C GNU General Public License for more details.
C
C You should have received a copy of the GNU General Public License
C along with this program. If not, see
C <http://www.gnu.org/licenses/>.
C
C PURPOSE
C
C To build and solve the least squares problem T*X = Kv, and
C estimate the matrices B and D of a linear time-invariant (LTI)
C state space model, using the solution X, and the singular
C value decomposition information and other intermediate results,
C provided by other routines.
C
C The matrix T is computed as a sum of Kronecker products,
C
C T = T + kron(Uf(:,(i-1)*m+1:i*m),N_i), for i = 1 : s,
C
C (with T initialized by zero), where Uf is the triangular
C factor of the QR factorization of the future input part (see
C SLICOT Library routine IB01ND), N_i is given by the i-th block
C row of the matrix
C
C [ Q_11 Q_12 ... Q_1,s-2 Q_1,s-1 Q_1s ] [ I_L 0 ]
C [ Q_12 Q_13 ... Q_1,s-1 Q_1s 0 ] [ ]
C N = [ Q_13 Q_14 ... Q_1s 0 0 ] * [ ],
C [ : : : : : ] [ ]
C [ Q_1s 0 ... 0 0 0 ] [ 0 GaL ]
C
C and where
C
C [ -L_1|1 ] [ M_i-1 - L_1|i ]
C Q_11 = [ ], Q_1i = [ ], i = 2:s,
C [ I_L - L_2|1 ] [ -L_2|i ]
C
C are (n+L)-by-L matrices, and GaL is built from the first n
C relevant singular vectors, GaL = Un(1:L(s-1),1:n), computed
C by IB01ND.
C
C The vector Kv is vec(K), with the matrix K defined by
C
C K = [ K_1 K_2 K_3 ... K_s ],
C
C where K_i = K(:,(i-1)*m+1:i*m), i = 1:s, is (n+L)-by-m.
C The given matrices are Uf, GaL, and
C
C [ L_1|1 ... L_1|s ]
C L = [ ], (n+L)-by-L*s,
C [ L_2|1 ... L_2|s ]
C
C M = [ M_1 ... M_s-1 ], n-by-L*(s-1), and
C K, (n+L)-by-m*s.
C
C Matrix M is the pseudoinverse of the matrix GaL, computed by
C SLICOT Library routine IB01PD.
C
C ARGUMENTS
C
C Mode Parameters
C
C JOB CHARACTER*1
C Specifies which of the matrices B and D should be
C computed, as follows:
C = 'B': compute the matrix B, but not the matrix D;
C = 'D': compute both matrices B and D.
C
C Input/Output Parameters
C
C NOBR (input) INTEGER
C The number of block rows, s, in the input and output
C Hankel matrices processed by other routines. NOBR > 1.
C
C N (input) INTEGER
C The order of the system. NOBR > N > 0.
C
C M (input) INTEGER
C The number of system inputs. M >= 0.
C
C L (input) INTEGER
C The number of system outputs. L > 0.
C
C UF (input/output) DOUBLE PRECISION array, dimension
C ( LDUF,M*NOBR )
C On entry, the leading M*NOBR-by-M*NOBR upper triangular
C part of this array must contain the upper triangular
C factor of the QR factorization of the future input part,
C as computed by SLICOT Library routine IB01ND.
C The strict lower triangle need not be set to zero.
C On exit, the leading M*NOBR-by-M*NOBR upper triangular
C part of this array is unchanged, and the strict lower
C triangle is set to zero.
C
C LDUF INTEGER
C The leading dimension of the array UF.
C LDUF >= MAX( 1, M*NOBR ).
C
C UN (input) DOUBLE PRECISION array, dimension ( LDUN,N )
C The leading L*(NOBR-1)-by-N part of this array must
C contain the matrix GaL, i.e., the leading part of the
C first N columns of the matrix Un of relevant singular
C vectors.
C
C LDUN INTEGER
C The leading dimension of the array UN.
C LDUN >= L*(NOBR-1).
C
C UL (input/output) DOUBLE PRECISION array, dimension
C ( LDUL,L*NOBR )
C On entry, the leading (N+L)-by-L*NOBR part of this array
C must contain the given matrix L.
C On exit, if M > 0, the leading (N+L)-by-L*NOBR part of
C this array is overwritten by the matrix
C [ Q_11 Q_12 ... Q_1,s-2 Q_1,s-1 Q_1s ].
C
C LDUL INTEGER
C The leading dimension of the array UL. LDUL >= N+L.
C
C PGAL (input) DOUBLE PRECISION array, dimension
C ( LDPGAL,L*(NOBR-1) )
C The leading N-by-L*(NOBR-1) part of this array must
C contain the pseudoinverse of the matrix GaL, computed by
C SLICOT Library routine IB01PD.
C
C LDPGAL INTEGER
C The leading dimension of the array PGAL. LDPGAL >= N.
C
C K (input) DOUBLE PRECISION array, dimension ( LDK,M*NOBR )
C The leading (N+L)-by-M*NOBR part of this array must
C contain the given matrix K.
C
C LDK INTEGER
C The leading dimension of the array K. LDK >= N+L.
C
C R (output) DOUBLE PRECISION array, dimension ( LDR,M*(N+L) )
C The leading (N+L)*M*NOBR-by-M*(N+L) part of this array
C contains details of the complete orthogonal factorization
C of the coefficient matrix T of the least squares problem
C which is solved for getting the system matrices B and D.
C
C LDR INTEGER
C The leading dimension of the array R.
C LDR >= MAX( 1, (N+L)*M*NOBR ).
C
C X (output) DOUBLE PRECISION array, dimension
C ( (N+L)*M*NOBR )
C The leading M*(N+L) elements of this array contain the
C least squares solution of the system T*X = Kv.
C The remaining elements are used as workspace (to store the
C corresponding part of the vector Kv = vec(K)).
C
C B (output) DOUBLE PRECISION array, dimension ( LDB,M )
C The leading N-by-M part of this array contains the system
C input matrix.
C
C LDB INTEGER
C The leading dimension of the array B. LDB >= N.
C
C D (output) DOUBLE PRECISION array, dimension ( LDD,M )
C If JOB = 'D', the leading L-by-M part of this array
C contains the system input-output matrix.
C If JOB = 'B', this array is not referenced.
C
C LDD INTEGER
C The leading dimension of the array D.
C LDD >= L, if JOB = 'D';
C LDD >= 1, if JOB = 'B'.
C
C Tolerances
C
C TOL DOUBLE PRECISION
C The tolerance to be used for estimating the rank of
C matrices. If the user sets TOL > 0, then the given value
C of TOL is used as a lower bound for the reciprocal
C condition number; an m-by-n matrix whose estimated
C condition number is less than 1/TOL is considered to
C be of full rank. If the user sets TOL <= 0, then an
C implicitly computed, default tolerance, defined by
C TOLDEF = m*n*EPS, is used instead, where EPS is the
C relative machine precision (see LAPACK Library routine
C DLAMCH).
C
C Workspace
C
C IWORK INTEGER array, dimension ( M*(N+L) )
C
C DWORK DOUBLE PRECISION array, dimension ( LDWORK )
C On exit, if INFO = 0, DWORK(1) returns the optimal value
C of LDWORK, and, if M > 0, DWORK(2) contains the
C reciprocal condition number of the triangular factor of
C the matrix T.
C On exit, if INFO = -26, DWORK(1) returns the minimum
C value of LDWORK.
C
C LDWORK INTEGER
C The length of the array DWORK.
C LDWORK >= MAX( (N+L)*(N+L), 4*M*(N+L)+1 ).
C For good performance, LDWORK should be larger.
C
C Warning Indicator
C
C IWARN INTEGER
C = 0: no warning;
C = 4: the least squares problem to be solved has a
C rank-deficient coefficient matrix.
C
C Error Indicator
C
C INFO INTEGER
C = 0: successful exit;
C < 0: if INFO = -i, the i-th argument had an illegal
C value.
C
C METHOD
C
C The matrix T is computed, evaluating the sum of Kronecker
C products, and then the linear system T*X = Kv is solved in a
C least squares sense. The matrices B and D are then directly
C obtained from the least squares solution.
C
C REFERENCES
C
C [1] Verhaegen M., and Dewilde, P.
C Subspace Model Identification. Part 1: The output-error
C state-space model identification class of algorithms.
C Int. J. Control, 56, pp. 1187-1210, 1992.
C
C [2] Van Overschee, P., and De Moor, B.
C N4SID: Two Subspace Algorithms for the Identification
C of Combined Deterministic-Stochastic Systems.
C Automatica, Vol.30, No.1, pp. 75-93, 1994.
C
C [3] Van Overschee, P.
C Subspace Identification : Theory - Implementation -
C Applications.
C Ph. D. Thesis, Department of Electrical Engineering,
C Katholieke Universiteit Leuven, Belgium, Feb. 1995.
C
C NUMERICAL ASPECTS
C
C The implemented method is numerically stable.
C
C CONTRIBUTOR
C
C V. Sima, Katholieke Universiteit Leuven, Feb. 2000.
C
C REVISIONS
C
C V. Sima, Katholieke Universiteit Leuven, Sep. 2001.
C
C KEYWORDS
C
C Identification methods; least squares solutions; multivariable
C systems; QR decomposition; singular value decomposition.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D0, ONE = 1.0D0 )
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( * )
C .. Local Scalars ..
DOUBLE PRECISION RCOND, TOLL
INTEGER I, IERR, J, JWORK, LDUN2, LNOBR, LP1, MAXWRK,
$ MINWRK, MKRON, MNOBR, NKRON, NP1, NPL, RANK
LOGICAL WITHB, WITHD
C .. External Functions ..
LOGICAL LSAME
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH, LSAME
C .. External Subroutines ..
EXTERNAL DGELSY, DGEMM, DLACPY, DLASET, DTRCON, MB01VD,
$ XERBLA
C .. Intrinsic Functions ..
INTRINSIC MAX
C .. Executable Statements ..
C
C Decode the scalar input parameters.
C
WITHD = LSAME( JOB, 'D' )
WITHB = LSAME( JOB, 'B' ) .OR. WITHD
MNOBR = M*NOBR
LNOBR = L*NOBR
LDUN2 = LNOBR - L
LP1 = L + 1
NP1 = N + 1
NPL = N + L
IWARN = 0
INFO = 0
C
C Check the scalar input parameters.
C
IF( .NOT.WITHB ) THEN
INFO = -1
ELSE IF( NOBR.LE.1 ) THEN
INFO = -2
ELSE IF( N.GE.NOBR .OR. N.LE.0 ) THEN
INFO = -3
ELSE IF( M.LT.0 ) THEN
INFO = -4
ELSE IF( L.LE.0 ) THEN
INFO = -5
ELSE IF( LDUF.LT.MAX( 1, MNOBR ) ) THEN
INFO = -7
ELSE IF( LDUN.LT.LDUN2 ) THEN
INFO = -9
ELSE IF( LDUL.LT.NPL ) THEN
INFO = -11
ELSE IF( LDPGAL.LT.N ) THEN
INFO = -13
ELSE IF( LDK.LT.NPL ) THEN
INFO = -15
ELSE IF( LDR.LT.MAX( 1, MNOBR*NPL ) ) THEN
INFO = -17
ELSE IF( LDB.LT.N ) THEN
INFO = -20
ELSE IF( LDD.LT.1 .OR. ( WITHD .AND. LDD.LT.L ) ) THEN
INFO = -22
ELSE
C
C Compute workspace.
C (Note: Comments in the code beginning "Workspace:" describe the
C minimal amount of workspace needed at that point in the code,
C as well as the preferred amount for good performance.
C NB refers to the optimal block size for the immediately
C following subroutine, as returned by ILAENV.)
C
MINWRK = MAX( NPL*NPL, 4*M*NPL + 1 )
C
IF ( LDWORK.LT.MINWRK ) THEN
INFO = -26
DWORK( 1 ) = MINWRK
END IF
END IF
C
C Return if there are illegal arguments.
C
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'IB01PX', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF( M.EQ.0 ) THEN
DWORK(1) = ONE
RETURN
END IF
C
C Construct the matrix [ Q_11 Q_12 ... Q_1,s-1 Q_1s ] in UL.
C
DO 20 J = 1, L
C
DO 10 I = 1, NPL
UL(I,J) = -UL(I,J)
10 CONTINUE
C
UL(N+J,J) = ONE + UL(N+J,J)
20 CONTINUE
C
DO 50 J = LP1, LNOBR
C
DO 30 I = 1, N
UL(I,J) = PGAL(I,J-L) - UL(I,J)
30 CONTINUE
C
DO 40 I = NP1, NPL
UL(I,J) = -UL(I,J)
40 CONTINUE
C
50 CONTINUE
C
C Compute the coefficient matrix T using Kronecker products.
C Workspace: (N+L)*(N+L).
C In the same loop, vectorize K in X.
C
CALL DLASET( 'Full', MNOBR*NPL, M*NPL, ZERO, ZERO, R, LDR )
CALL DLASET( 'Lower', MNOBR-1, MNOBR-1, ZERO, ZERO, UF(2,1),
$ LDUF )
JWORK = NPL*L + 1
C
DO 60 I = 1, NOBR
CALL DLACPY( 'Full', NPL, L, UL(1,(I-1)*L+1), LDUL, DWORK,
$ NPL )
IF ( I.LT.NOBR ) THEN
CALL DGEMM ( 'NoTranspose', 'NoTranspose', NPL, N,
$ L*(NOBR-I), ONE, UL(1,I*L+1), LDUL, UN, LDUN,
$ ZERO, DWORK(JWORK), NPL )
ELSE
CALL DLASET( 'Full', NPL, N, ZERO, ZERO, DWORK(JWORK), NPL )
END IF
CALL MB01VD( 'NoTranspose', 'NoTranspose', MNOBR, M, NPL,
$ NPL, ONE, ONE, UF(1,(I-1)*M+1), LDUF, DWORK,
$ NPL, R, LDR, MKRON, NKRON, IERR )
CALL DLACPY( 'Full', NPL, M, K(1,(I-1)*M+1), LDK,
$ X((I-1)*NKRON+1), NPL )
60 CONTINUE
C
C Compute the tolerance.
C
TOLL = TOL
IF( TOLL.LE.ZERO )
$ TOLL = MKRON*NKRON*DLAMCH( 'Precision' )
C
C Solve the least square problem T*X = vec(K).
C Workspace: need 4*M*(N+L)+1;
C prefer 3*M*(N+L)+(M*(N+L)+1)*NB.
C
DO 70 I = 1, NKRON
IWORK(I) = 0
70 CONTINUE
C
CALL DGELSY( MKRON, NKRON, 1, R, LDR, X, MKRON, IWORK, TOLL, RANK,
$ DWORK, LDWORK, IERR )
MAXWRK = DWORK(1)
C
C Compute the reciprocal of the condition number of the triangular
C factor R of T.
C Workspace: need 3*M*(N+L).
C
CALL DTRCON( '1-norm', 'Upper', 'NonUnit', NKRON, R, LDR, RCOND,
$ DWORK, IWORK, IERR )
C
IF ( RANK.LT.NKRON ) THEN
C
C The least squares problem is rank-deficient.
C
IWARN = 4
END IF
C
C Construct the matrix D, if needed.
C
IF ( WITHD )
$ CALL DLACPY( 'Full', L, M, X, NPL, D, LDD )
C
C Construct the matrix B.
C
CALL DLACPY( 'Full', N, M, X(LP1), NPL, B, LDB )
C
C Return optimal workspace in DWORK(1) and reciprocal condition
C number in DWORK(2).
C
DWORK(1) = MAX( MINWRK, MAXWRK )
DWORK(2) = RCOND
C
RETURN
C
C *** Last line of IB01PX ***
END
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