SUBROUTINE SB10TD( N, M, NP, NCON, NMEAS, D, LDD, TU, LDTU, TY,
$ LDTY, AK, LDAK, BK, LDBK, CK, LDCK, DK, LDDK,
$ RCOND, TOL, IWORK, DWORK, LDWORK, INFO )
C
C PURPOSE
C
C To compute the matrices of the H2 optimal discrete-time controller
C
C | AK | BK |
C K = |----|----|,
C | CK | DK |
C
C from the matrices of the controller for the normalized system,
C as determined by the SLICOT Library routine SB10SD.
C
C ARGUMENTS
C
C Input/Output Parameters
C
C N (input) INTEGER
C The order of the system. N >= 0.
C
C M (input) INTEGER
C The column size of the matrix B. M >= 0.
C
C NP (input) INTEGER
C The row size of the matrix C. NP >= 0.
C
C NCON (input) INTEGER
C The number of control inputs (M2). M >= NCON >= 0.
C NP-NMEAS >= NCON.
C
C NMEAS (input) INTEGER
C The number of measurements (NP2). NP >= NMEAS >= 0.
C M-NCON >= NMEAS.
C
C D (input) DOUBLE PRECISION array, dimension (LDD,M)
C The leading NP-by-M part of this array must contain the
C system input/output matrix D. Only the trailing
C NMEAS-by-NCON submatrix D22 is used.
C
C LDD INTEGER
C The leading dimension of the array D. LDD >= max(1,NP).
C
C TU (input) DOUBLE PRECISION array, dimension (LDTU,M2)
C The leading M2-by-M2 part of this array must contain the
C control transformation matrix TU, as obtained by the
C SLICOT Library routine SB10PD.
C
C LDTU INTEGER
C The leading dimension of the array TU. LDTU >= max(1,M2).
C
C TY (input) DOUBLE PRECISION array, dimension (LDTY,NP2)
C The leading NP2-by-NP2 part of this array must contain the
C measurement transformation matrix TY, as obtained by the
C SLICOT Library routine SB10PD.
C
C LDTY INTEGER
C The leading dimension of the array TY.
C LDTY >= max(1,NP2).
C
C AK (input/output) DOUBLE PRECISION array, dimension (LDAK,N)
C On entry, the leading N-by-N part of this array must
C contain controller state matrix for the normalized system
C as obtained by the SLICOT Library routine SB10SD.
C On exit, the leading N-by-N part of this array contains
C controller state matrix AK.
C
C LDAK INTEGER
C The leading dimension of the array AK. LDAK >= max(1,N).
C
C BK (input/output) DOUBLE PRECISION array, dimension
C (LDBK,NMEAS)
C On entry, the leading N-by-NMEAS part of this array must
C contain controller input matrix for the normalized system
C as obtained by the SLICOT Library routine SB10SD.
C On exit, the leading N-by-NMEAS part of this array
C contains controller input matrix BK.
C
C LDBK INTEGER
C The leading dimension of the array BK. LDBK >= max(1,N).
C
C CK (input/output) DOUBLE PRECISION array, dimension (LDCK,N)
C On entry, the leading NCON-by-N part of this array must
C contain controller output matrix for the normalized
C system as obtained by the SLICOT Library routine SB10SD.
C On exit, the leading NCON-by-N part of this array contains
C controller output matrix CK.
C
C LDCK INTEGER
C The leading dimension of the array CK.
C LDCK >= max(1,NCON).
C
C DK (input/output) DOUBLE PRECISION array, dimension
C (LDDK,NMEAS)
C On entry, the leading NCON-by-NMEAS part of this array
C must contain controller matrix DK for the normalized
C system as obtained by the SLICOT Library routine SB10SD.
C On exit, the leading NCON-by-NMEAS part of this array
C contains controller input/output matrix DK.
C
C LDDK INTEGER
C The leading dimension of the array DK.
C LDDK >= max(1,NCON).
C
C RCOND (output) DOUBLE PRECISION
C RCOND contains an estimate of the reciprocal condition
C number of the matrix Im2 + DKHAT*D22 which must be
C inverted in the computation of the controller.
C
C Tolerances
C
C TOL DOUBLE PRECISION
C Tolerance used in determining the nonsingularity of the
C matrix which must be inverted. If TOL <= 0, then a default
C value equal to sqrt(EPS) is used, where EPS is the
C relative machine precision.
C
C Workspace
C
C IWORK INTEGER array, dimension (2*M2)
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C
C LDWORK INTEGER
C The dimension of the array DWORK.
C LDWORK >= max(N*M2,N*NP2,M2*NP2,M2*M2+4*M2).
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 = 1: if the matrix Im2 + DKHAT*D22 is singular, or the
C estimated condition number is larger than or equal
C to 1/TOL.
C
C METHOD
C
C The routine implements the formulas given in [1].
C
C REFERENCES
C
C [1] Zhou, K., Doyle, J.C., and Glover, K.
C Robust and Optimal Control.
C Prentice-Hall, Upper Saddle River, NJ, 1996.
C
C [2] Petkov, P.Hr., Gu, D.W., and Konstantinov, M.M.
C Fortran 77 routines for Hinf and H2 design of linear
C discrete-time control systems.
C Report 99-8, Department of Engineering, Leicester University,
C April 1999.
C
C NUMERICAL ASPECTS
C
C The accuracy of the result depends on the condition numbers of the
C input and output transformations and of the matrix Im2 +
C DKHAT*D22.
C
C CONTRIBUTORS
C
C P.Hr. Petkov, D.W. Gu and M.M. Konstantinov, April 1999.
C
C REVISIONS
C
C V. Sima, Research Institute for Informatics, Bucharest, May 1999,
C Jan. 2000.
C
C KEYWORDS
C
C Algebraic Riccati equation, H2 optimal control, LQG, LQR, optimal
C regulator, robust control.
C
C ******************************************************************
C
C .. Parameters ..
DOUBLE PRECISION ZERO, ONE
PARAMETER ( ZERO = 0.0D+0, ONE = 1.0D+0 )
C ..
C .. Scalar Arguments ..
INTEGER INFO, LDAK, LDBK, LDCK, LDD, LDDK, LDTU, LDTY,
$ LDWORK, M, N, NCON, NMEAS, NP
DOUBLE PRECISION RCOND, TOL
C ..
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION AK( LDAK, * ), BK( LDBK, * ), CK( LDCK, * ),
$ D( LDD, * ), DK( LDDK, * ), DWORK( * ),
$ TU( LDTU, * ), TY( LDTY, * )
C ..
C .. Local Scalars ..
INTEGER INFO2, IWRK, M1, M2, MINWRK, NP1, NP2
DOUBLE PRECISION ANORM, TOLL
C ..
C .. External Functions
DOUBLE PRECISION DLAMCH, DLANGE
EXTERNAL DLAMCH, DLANGE
C ..
C .. External Subroutines ..
EXTERNAL DGECON, DGEMM, DGETRF, DGETRS, DLACPY, DLASET,
$ XERBLA
C ..
C .. Intrinsic Functions ..
INTRINSIC MAX, SQRT
C ..
C .. Executable Statements ..
C
C Decode and Test input parameters.
C
M1 = M - NCON
M2 = NCON
NP1 = NP - NMEAS
NP2 = NMEAS
C
INFO = 0
IF( N.LT.0 ) THEN
INFO = -1
ELSE IF( M.LT.0 ) THEN
INFO = -2
ELSE IF( NP.LT.0 ) THEN
INFO = -3
ELSE IF( NCON.LT.0 .OR. M1.LT.0 .OR. M2.GT.NP1 ) THEN
INFO = -4
ELSE IF( NMEAS.LT.0 .OR. NP1.LT.0 .OR. NP2.GT.M1 ) THEN
INFO = -5
ELSE IF( LDD.LT.MAX( 1, NP ) ) THEN
INFO = -7
ELSE IF( LDTU.LT.MAX( 1, M2 ) ) THEN
INFO = -9
ELSE IF( LDTY.LT.MAX( 1, NP2 ) ) THEN
INFO = -11
ELSE IF( LDAK.LT.MAX( 1, N ) ) THEN
INFO = -13
ELSE IF( LDBK.LT.MAX( 1, N ) ) THEN
INFO = -15
ELSE IF( LDCK.LT.MAX( 1, M2 ) ) THEN
INFO = -17
ELSE IF( LDDK.LT.MAX( 1, M2 ) ) THEN
INFO = -19
ELSE
C
C Compute workspace.
C
MINWRK = MAX ( N*M2, N*NP2, M2*NP2, M2*( M2 + 4 ) )
IF( LDWORK.LT.MINWRK )
$ INFO = -24
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SB10TD', -INFO )
RETURN
END IF
C
C Quick return if possible.
C
IF( N.EQ.0 .OR. M.EQ.0 .OR. NP.EQ.0 .OR. M1.EQ.0 .OR. M2.EQ.0
$ .OR. NP1.EQ.0 .OR. NP2.EQ.0 ) THEN
RCOND = ONE
RETURN
END IF
C
TOLL = TOL
IF( TOLL.LE.ZERO ) THEN
C
C Set the default value of the tolerance for nonsingularity test.
C
TOLL = SQRT( DLAMCH( 'Epsilon' ) )
END IF
C
C Find BKHAT .
C
CALL DGEMM( 'N', 'N', N, NP2, NP2, ONE, BK, LDBK, TY, LDTY, ZERO,
$ DWORK, N )
CALL DLACPY ('Full', N, NP2, DWORK, N, BK, LDBK )
C
C Find CKHAT .
C
CALL DGEMM( 'N', 'N', M2, N, M2, ONE, TU, LDTU, CK, LDCK, ZERO,
$ DWORK, M2 )
CALL DLACPY ('Full', M2, N, DWORK, M2, CK, LDCK )
C
C Compute DKHAT .
C
CALL DGEMM( 'N', 'N', M2, NP2, M2, ONE, TU, LDTU, DK, LDDK, ZERO,
$ DWORK, M2 )
CALL DGEMM( 'N', 'N', M2, NP2, NP2, ONE, DWORK, M2, TY, LDTY,
$ ZERO, DK, LDDK )
C
C Compute Im2 + DKHAT*D22 .
C
IWRK = M2*M2 + 1
CALL DLASET( 'Full', M2, M2, ZERO, ONE, DWORK, M2 )
CALL DGEMM( 'N', 'N', M2, M2, NP2, ONE, DK, LDDK,
$ D( NP1+1, M1+1 ), LDD, ONE, DWORK, M2 )
ANORM = DLANGE( '1', M2, M2, DWORK, M2, DWORK( IWRK ) )
CALL DGETRF( M2, M2, DWORK, M2, IWORK, INFO2 )
IF( INFO2.GT.0 ) THEN
INFO = 1
RETURN
END IF
CALL DGECON( '1', M2, DWORK, M2, ANORM, RCOND, DWORK( IWRK ),
$ IWORK( M2+1 ), INFO2 )
C
C Return if the matrix is singular to working precision.
C
IF( RCOND.LT.TOLL ) THEN
INFO = 1
RETURN
END IF
C
C Compute CK .
C
CALL DGETRS( 'N', M2, N, DWORK, M2, IWORK, CK, LDCK, INFO2 )
C
C Compute DK .
C
CALL DGETRS( 'N', M2, NP2, DWORK, M2, IWORK, DK, LDDK, INFO2 )
C
C Compute AK .
C
CALL DGEMM( 'N', 'N', N, M2, NP2, ONE, BK, LDBK, D( NP1+1, M1+1 ),
$ LDD, ZERO, DWORK, N )
CALL DGEMM( 'N', 'N', N, N, M2, -ONE, DWORK, N, CK, LDCK, ONE, AK,
$ LDAK )
C
C Compute BK .
C
CALL DGEMM( 'N', 'N', N, NP2, M2, -ONE, DWORK, N, DK, LDDK,
$ ONE, BK, LDBK )
RETURN
C *** Last line of SB10TD ***
END