SUBROUTINE SB10VD( N, M, NP, NCON, NMEAS, A, LDA, B, LDB, C, LDC,
$ F, LDF, H, LDH, X, LDX, Y, LDY, XYCOND, IWORK,
$ DWORK, LDWORK, BWORK, INFO )
C
C PURPOSE
C
C To compute the state feedback and the output injection
C matrices for an H2 optimal n-state controller for the system
C
C | A | B1 B2 | | A | B |
C P = |----|---------| = |---|---|
C | C1 | 0 D12 | | C | D |
C | C2 | D21 D22 |
C
C where B2 has as column size the number of control inputs (NCON)
C and C2 has as row size the number of measurements (NMEAS) being
C provided to the controller.
C
C It is assumed that
C
C (A1) (A,B2) is stabilizable and (C2,A) is detectable,
C
C (A2) D12 is full column rank with D12 = | 0 | and D21 is
C | I |
C full row rank with D21 = | 0 I | as obtained by the
C SLICOT Library routine SB10UD. Matrix D is not used
C explicitly.
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 A (input) DOUBLE PRECISION array, dimension (LDA,N)
C The leading N-by-N part of this array must contain the
C system state matrix A.
C
C LDA INTEGER
C The leading dimension of the array A. LDA >= max(1,N).
C
C B (input) DOUBLE PRECISION array, dimension (LDB,M)
C The leading N-by-M part of this array must contain the
C system input matrix B.
C
C LDB INTEGER
C The leading dimension of the array B. LDB >= max(1,N).
C
C C (input) DOUBLE PRECISION array, dimension (LDC,N)
C The leading NP-by-N part of this array must contain the
C system output matrix C.
C
C LDC INTEGER
C The leading dimension of the array C. LDC >= max(1,NP).
C
C F (output) DOUBLE PRECISION array, dimension (LDF,N)
C The leading NCON-by-N part of this array contains the
C state feedback matrix F.
C
C LDF INTEGER
C The leading dimension of the array F. LDF >= max(1,NCON).
C
C H (output) DOUBLE PRECISION array, dimension (LDH,NMEAS)
C The leading N-by-NMEAS part of this array contains the
C output injection matrix H.
C
C LDH INTEGER
C The leading dimension of the array H. LDH >= max(1,N).
C
C X (output) DOUBLE PRECISION array, dimension (LDX,N)
C The leading N-by-N part of this array contains the matrix
C X, solution of the X-Riccati equation.
C
C LDX INTEGER
C The leading dimension of the array X. LDX >= max(1,N).
C
C Y (output) DOUBLE PRECISION array, dimension (LDY,N)
C The leading N-by-N part of this array contains the matrix
C Y, solution of the Y-Riccati equation.
C
C LDY INTEGER
C The leading dimension of the array Y. LDY >= max(1,N).
C
C XYCOND (output) DOUBLE PRECISION array, dimension (2)
C XYCOND(1) contains an estimate of the reciprocal condition
C number of the X-Riccati equation;
C XYCOND(2) contains an estimate of the reciprocal condition
C number of the Y-Riccati equation.
C
C Workspace
C
C IWORK INTEGER array, dimension (max(2*N,N*N))
C
C DWORK DOUBLE PRECISION array, dimension (LDWORK)
C On exit, if INFO = 0, DWORK(1) contains the optimal
C LDWORK.
C
C LDWORK INTEGER
C The dimension of the array DWORK.
C LDWORK >= 13*N*N + 12*N + 5.
C For good performance, LDWORK must generally be larger.
C
C BWORK LOGICAL array, dimension (2*N)
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 X-Riccati equation was not solved
C successfully;
C = 2: if the Y-Riccati equation was not solved
C successfully.
C
C METHOD
C
C The routine implements the formulas given in [1], [2]. The X-
C and Y-Riccati equations are solved with condition and accuracy
C estimates [3].
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] Balas, G.J., Doyle, J.C., Glover, K., Packard, A., and
C Smith, R.
C mu-Analysis and Synthesis Toolbox.
C The MathWorks Inc., Natick, Mass., 1995.
C
C [3] Petkov, P.Hr., Konstantinov, M.M., and Mehrmann, V.
C DGRSVX and DMSRIC: Fortan 77 subroutines for solving
C continuous-time matrix algebraic Riccati equations with
C condition and accuracy estimates.
C Preprint SFB393/98-16, Fak. f. Mathematik, Tech. Univ.
C Chemnitz, May 1998.
C
C NUMERICAL ASPECTS
C
C The precision of the solution of the matrix Riccati equations
C can be controlled by the values of the condition numbers
C XYCOND(1) and XYCOND(2) of these equations.
C
C FURTHER COMMENTS
C
C The Riccati equations are solved by the Schur approach
C implementing condition and accuracy estimates.
C
C CONTRIBUTORS
C
C P.Hr. Petkov, D.W. Gu and M.M. Konstantinov, October 1998.
C
C REVISIONS
C
C V. Sima, Research Institute for Informatics, Bucharest, May 1999.
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, LDA, LDB, LDC, LDF, LDH, LDWORK, LDX,
$ LDY, M, N, NCON, NMEAS, NP
C ..
C .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ DWORK( * ), F( LDF, * ), H( LDH, * ),
$ X( LDX, * ), XYCOND( 2 ), Y( LDY, * )
C ..
C .. Local Scalars ..
INTEGER INFO2, IWG, IWI, IWQ, IWR, IWRK, IWS, IWT, IWV,
$ LWAMAX, M1, M2, MINWRK, N2, ND1, ND2, NP1, NP2
DOUBLE PRECISION FERR, SEP
C ..
C .. External Functions ..
C
DOUBLE PRECISION DLANSY
EXTERNAL DLANSY
C ..
C .. External Subroutines ..
EXTERNAL DGEMM, DLACPY, DLASET, DSYRK, SB02RD, XERBLA
C ..
C .. Intrinsic Functions ..
INTRINSIC DBLE, INT, MAX
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( LDA.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -9
ELSE IF( LDC.LT.MAX( 1, NP ) ) THEN
INFO = -11
ELSE IF( LDF.LT.MAX( 1, NCON ) ) THEN
INFO = -13
ELSE IF( LDH.LT.MAX( 1, N ) ) THEN
INFO = -15
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -17
ELSE IF( LDY.LT.MAX( 1, N ) ) THEN
INFO = -19
ELSE
C
C Compute workspace.
C
MINWRK = 13*N*N + 12*N + 5
IF( LDWORK.LT.MINWRK )
$ INFO = -23
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SB10VD', -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
DWORK( 1 ) = ONE
XYCOND( 1 ) = ONE
XYCOND( 2 ) = ONE
RETURN
END IF
C
ND1 = NP1 - M2
ND2 = M1 - NP2
N2 = 2*N
C
C Workspace usage.
C
IWQ = N*N + 1
IWG = IWQ + N*N
IWT = IWG + N*N
IWV = IWT + N*N
IWR = IWV + N*N
IWI = IWR + N2
IWS = IWI + N2
IWRK = IWS + 4*N*N
C
C Compute Ax = A - B2*D12'*C1 .
C
CALL DLACPY ('Full', N, N, A, LDA, DWORK, N )
CALL DGEMM( 'N', 'N', N, N, M2, -ONE, B( 1, M1+1 ), LDB,
$ C( ND1+1, 1), LDC, ONE, DWORK, N )
C
C Compute Cx = C1'*C1 - C1'*D12*D12'*C1 .
C
IF( ND1.GT.0 ) THEN
CALL DSYRK( 'L', 'T', N, ND1, ONE, C, LDC, ZERO, DWORK( IWQ ),
$ N )
ELSE
CALL DLASET( 'L', N, N, ZERO, ZERO, DWORK( IWQ ), N )
END IF
C
C Compute Dx = B2*B2' .
C
CALL DSYRK( 'L', 'N', N, M2, ONE, B( 1, M1+1 ), LDB, ZERO,
$ DWORK( IWG ), N )
C
C Solution of the Riccati equation Ax'*X + X*Ax + Cx - X*Dx*X = 0 .
C Workspace: need 13*N*N + 12*N + 5;
C prefer larger.
C
CALL SB02RD( 'All', 'Continuous', 'NotUsed', 'NoTranspose',
$ 'Lower', 'GeneralScaling', 'Stable', 'NotFactored',
$ 'Original', N, DWORK, N, DWORK( IWT ), N,
$ DWORK( IWV ), N, DWORK( IWG ), N, DWORK( IWQ ), N,
$ X, LDX, SEP, XYCOND( 1 ), FERR, DWORK( IWR ),
$ DWORK( IWI ), DWORK( IWS ), N2, IWORK,
$ DWORK( IWRK ), LDWORK-IWRK+1, BWORK, INFO2 )
IF( INFO2.GT.0 ) THEN
INFO = 1
RETURN
END IF
C
LWAMAX = INT( DWORK( IWRK ) ) + IWRK - 1
C
C Compute F = -D12'*C1 - B2'*X .
C
CALL DLACPY( 'Full', M2, N, C( ND1+1, 1 ), LDC, F, LDF )
CALL DGEMM( 'T', 'N', M2, N, N, -ONE, B( 1, M1+1 ), LDB, X, LDX,
$ -ONE, F, LDF )
C
C Compute Ay = A - B1*D21'*C2 .
C
CALL DLACPY( 'Full', N, N, A, LDA, DWORK, N )
CALL DGEMM( 'N', 'N', N, N, NP2, -ONE, B( 1, ND2+1 ), LDB,
$ C( NP1+1, 1 ), LDC, ONE, DWORK, N )
C
C Compute Cy = B1*B1' - B1*D21'*D21*B1' .
C
IF( ND2.GT.0 ) THEN
CALL DSYRK( 'U', 'N', N, ND2, ONE, B, LDB, ZERO, DWORK( IWQ ),
$ N )
ELSE
CALL DLASET( 'U', N, N, ZERO, ZERO, DWORK( IWQ ), N )
END IF
C
C Compute Dy = C2'*C2 .
C
CALL DSYRK( 'U', 'T', N, NP2, ONE, C( NP1+1, 1 ), LDC, ZERO,
$ DWORK( IWG ), N )
C
C Solution of the Riccati equation Ay*Y + Y*Ay' + Cy - Y*Dy*Y = 0 .
C Workspace: need 13*N*N + 12*N + 5;
C prefer larger.
C
CALL SB02RD( 'All', 'Continuous', 'NotUsed', 'Transpose',
$ 'Upper', 'GeneralScaling', 'Stable', 'NotFactored',
$ 'Original', N, DWORK, N, DWORK( IWT ), N,
$ DWORK( IWV ), N, DWORK( IWG ), N, DWORK( IWQ ), N,
$ Y, LDY, SEP, XYCOND( 2 ), FERR, DWORK( IWR ),
$ DWORK( IWI ), DWORK( IWS ), N2, IWORK,
$ DWORK( IWRK ), LDWORK-IWRK+1, BWORK, INFO2 )
IF( INFO2.GT.0 ) THEN
INFO = 2
RETURN
END IF
C
LWAMAX = MAX( INT( DWORK( IWRK ) ) + IWRK - 1, LWAMAX )
C
C Compute H = -B1*D21' - Y*C2' .
C
CALL DLACPY( 'Full', N, NP2, B( 1, ND2+1 ), LDB, H, LDH )
CALL DGEMM( 'N', 'T', N, NP2, N, -ONE, Y, LDY, C( NP1+1, 1 ), LDC,
$ -ONE, H, LDH )
C
DWORK( 1 ) = DBLE( LWAMAX )
RETURN
C *** Last line of SB10VD ***
END