SUBROUTINE SB10FD( N, M, NP, NCON, NMEAS, GAMMA, A, LDA, B, LDB,
$ C, LDC, D, LDD, AK, LDAK, BK, LDBK, CK, LDCK,
$ DK, LDDK, RCOND, TOL, IWORK, DWORK, LDWORK,
$ BWORK, INFO )
C
C PURPOSE
C
C To compute the matrices of an H-infinity (sub)optimal n-state
C controller
C
C | AK | BK |
C K = |----|----|,
C | CK | DK |
C
C using modified Glover's and Doyle's 1988 formulas, for the system
C
C | A | B1 B2 | | A | B |
C P = |----|---------| = |---|---|
C | C1 | D11 D12 | | C | D |
C | C2 | D21 D22 |
C
C and for a given value of gamma, where B2 has as column size the
C number of control inputs (NCON) and C2 has as row size the number
C of measurements (NMEAS) being 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 and D21 is full row rank,
C
C (A3) | A-j*omega*I B2 | has full column rank for all omega,
C | C1 D12 |
C
C (A4) | A-j*omega*I B1 | has full row rank for all omega.
C | C2 D21 |
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 GAMMA (input) DOUBLE PRECISION
C The value of gamma. It is assumed that gamma is
C sufficiently large so that the controller is admissible.
C GAMMA >= 0.
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 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.
C
C LDD INTEGER
C The leading dimension of the array D. LDD >= max(1,NP).
C
C AK (output) DOUBLE PRECISION array, dimension (LDAK,N)
C The leading N-by-N part of this array contains the
C controller state matrix AK.
C
C LDAK INTEGER
C The leading dimension of the array AK. LDAK >= max(1,N).
C
C BK (output) DOUBLE PRECISION array, dimension (LDBK,NMEAS)
C The leading N-by-NMEAS part of this array contains the
C controller input matrix BK.
C
C LDBK INTEGER
C The leading dimension of the array BK. LDBK >= max(1,N).
C
C CK (output) DOUBLE PRECISION array, dimension (LDCK,N)
C The leading NCON-by-N part of this array contains the
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 (output) DOUBLE PRECISION array, dimension (LDDK,NMEAS)
C The leading NCON-by-NMEAS part of this array contains the
C 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 array, dimension (4)
C RCOND(1) contains the reciprocal condition number of the
C control transformation matrix;
C RCOND(2) contains the reciprocal condition number of the
C measurement transformation matrix;
C RCOND(3) contains an estimate of the reciprocal condition
C number of the X-Riccati equation;
C RCOND(4) contains an estimate of the reciprocal condition
C number of the Y-Riccati equation.
C
C Tolerances
C
C TOL DOUBLE PRECISION
C Tolerance used for controlling the accuracy of the applied
C transformations for computing the normalized form in
C SLICOT Library routine SB10PD. Transformation matrices
C whose reciprocal condition numbers are less than TOL are
C not allowed. If TOL <= 0, then a default value equal to
C sqrt(EPS) is used, where EPS is the relative machine
C precision.
C
C Workspace
C
C IWORK INTEGER array, dimension (LIWORK), where
C LIWORK = max(2*max(N,M-NCON,NP-NMEAS,NCON),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 >= N*M + NP*(N+M) + M2*M2 + NP2*NP2 +
C max(1,LW1,LW2,LW3,LW4,LW5,LW6), where
C LW1 = (N+NP1+1)*(N+M2) + max(3*(N+M2)+N+NP1,5*(N+M2)),
C LW2 = (N+NP2)*(N+M1+1) + max(3*(N+NP2)+N+M1,5*(N+NP2)),
C LW3 = M2 + NP1*NP1 + max(NP1*max(N,M1),3*M2+NP1,5*M2),
C LW4 = NP2 + M1*M1 + max(max(N,NP1)*M1,3*NP2+M1,5*NP2),
C LW5 = 2*N*N + N*(M+NP) +
C max(1,M*M + max(2*M1,3*N*N+max(N*M,10*N*N+12*N+5)),
C NP*NP + max(2*NP1,3*N*N +
C max(N*NP,10*N*N+12*N+5))),
C LW6 = 2*N*N + N*(M+NP) +
C max(1, M2*NP2 + NP2*NP2 + M2*M2 +
C max(D1*D1 + max(2*D1, (D1+D2)*NP2),
C D2*D2 + max(2*D2, D2*M2), 3*N,
C N*(2*NP2 + M2) +
C max(2*N*M2, M2*NP2 +
C max(M2*M2+3*M2, NP2*(2*NP2+
C M2+max(NP2,N)))))),
C with D1 = NP1 - M2, D2 = M1 - NP2,
C NP1 = NP - NP2, M1 = M - M2.
C For good performance, LDWORK must generally be larger.
C Denoting Q = max(M1,M2,NP1,NP2), an upper bound is
C 2*Q*(3*Q+2*N)+max(1,(N+Q)*(N+Q+6),Q*(Q+max(N,Q,5)+1),
C 2*N*(N+2*Q)+max(1,4*Q*Q+
C max(2*Q,3*N*N+max(2*N*Q,10*N*N+12*N+5)),
C Q*(3*N+3*Q+max(2*N,4*Q+max(N,Q))))).
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 matrix | A-j*omega*I B2 | had not full
C | C1 D12 |
C column rank in respect to the tolerance EPS;
C = 2: if the matrix | A-j*omega*I B1 | had not full row
C | C2 D21 |
C rank in respect to the tolerance EPS;
C = 3: if the matrix D12 had not full column rank in
C respect to the tolerance TOL;
C = 4: if the matrix D21 had not full row rank in respect
C to the tolerance TOL;
C = 5: if the singular value decomposition (SVD) algorithm
C did not converge (when computing the SVD of one of
C the matrices |A B2 |, |A B1 |, D12 or D21).
C |C1 D12| |C2 D21|
C = 6: if the controller is not admissible (too small value
C of gamma);
C = 7: if the X-Riccati equation was not solved
C successfully (the controller is not admissible or
C there are numerical difficulties);
C = 8: if the Y-Riccati equation was not solved
C successfully (the controller is not admissible or
C there are numerical difficulties);
C = 9: if the determinant of Im2 + Tu*D11HAT*Ty*D22 is
C zero [3].
C
C METHOD
C
C The routine implements the Glover's and Doyle's 1988 formulas [1],
C [2] modified to improve the efficiency as described in [3].
C
C REFERENCES
C
C [1] Glover, K. and Doyle, J.C.
C State-space formulae for all stabilizing controllers that
C satisfy an Hinf norm bound and relations to risk sensitivity.
C Systems and Control Letters, vol. 11, pp. 167-172, 1988.
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., Gu, D.W., and Konstantinov, M.M.
C Fortran 77 routines for Hinf and H2 design of continuous-time
C linear control systems.
C Rep. 98-14, Department of Engineering, Leicester University,
C Leicester, U.K., 1998.
C
C NUMERICAL ASPECTS
C
C The accuracy of the result depends on the condition numbers of the
C input and output transformations and on the condition numbers of
C the two Riccati equations, as given by the values of RCOND(1),
C RCOND(2), RCOND(3) and RCOND(4), respectively.
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 Sept. 1999, Feb. 2000.
C
C KEYWORDS
C
C Algebraic Riccati equation, H-infinity optimal control, robust
C 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, LDAK, LDB, LDBK, LDC, LDCK, LDD,
$ LDDK, LDWORK, M, N, NCON, NMEAS, NP
DOUBLE PRECISION GAMMA, TOL
C ..
C .. Array Arguments ..
LOGICAL BWORK( * )
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), AK( LDAK, * ), B( LDB, * ),
$ BK( LDBK, * ), C( LDC, * ), CK( LDCK, * ),
$ D( LDD, * ), DK( LDDK, * ), DWORK( * ),
$ RCOND( 4 )
C ..
C .. Local Scalars ..
INTEGER INFO2, IWC, IWD, IWF, IWH, IWRK, IWTU, IWTY,
$ IWX, IWY, LW1, LW2, LW3, LW4, LW5, LW6,
$ LWAMAX, M1, M2, MINWRK, ND1, ND2, NP1, NP2
DOUBLE PRECISION TOLL
C ..
C .. External Functions ..
DOUBLE PRECISION DLAMCH
EXTERNAL DLAMCH
C ..
C .. External Subroutines ..
EXTERNAL DLACPY, SB10PD, SB10QD, SB10RD, XERBLA
C ..
C .. Intrinsic Functions ..
INTRINSIC DBLE, INT, 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( GAMMA.LT.ZERO ) THEN
INFO = -6
ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
INFO = -8
ELSE IF( LDB.LT.MAX( 1, N ) ) THEN
INFO = -10
ELSE IF( LDC.LT.MAX( 1, NP ) ) THEN
INFO = -12
ELSE IF( LDD.LT.MAX( 1, NP ) ) THEN
INFO = -14
ELSE IF( LDAK.LT.MAX( 1, N ) ) THEN
INFO = -16
ELSE IF( LDBK.LT.MAX( 1, N ) ) THEN
INFO = -18
ELSE IF( LDCK.LT.MAX( 1, M2 ) ) THEN
INFO = -20
ELSE IF( LDDK.LT.MAX( 1, M2 ) ) THEN
INFO = -22
ELSE
C
C Compute workspace.
C
ND1 = NP1 - M2
ND2 = M1 - NP2
LW1 = ( N + NP1 + 1 )*( N + M2 ) + MAX( 3*( N + M2 ) + N + NP1,
$ 5*( N + M2 ) )
LW2 = ( N + NP2 )*( N + M1 + 1 ) + MAX( 3*( N + NP2 ) + N +
$ M1, 5*( N + NP2 ) )
LW3 = M2 + NP1*NP1 + MAX( NP1*MAX( N, M1 ), 3*M2 + NP1, 5*M2 )
LW4 = NP2 + M1*M1 + MAX( MAX( N, NP1 )*M1, 3*NP2 + M1, 5*NP2 )
LW5 = 2*N*N + N*( M + NP ) +
$ MAX( 1, M*M + MAX( 2*M1, 3*N*N +
$ MAX( N*M, 10*N*N + 12*N + 5 ) ),
$ NP*NP + MAX( 2*NP1, 3*N*N +
$ MAX( N*NP, 10*N*N + 12*N + 5 ) ) )
LW6 = 2*N*N + N*( M + NP ) +
$ MAX( 1, M2*NP2 + NP2*NP2 + M2*M2 +
$ MAX( ND1*ND1 + MAX( 2*ND1, ( ND1 + ND2 )*NP2 ),
$ ND2*ND2 + MAX( 2*ND2, ND2*M2 ), 3*N,
$ N*( 2*NP2 + M2 ) +
$ MAX( 2*N*M2, M2*NP2 +
$ MAX( M2*M2 + 3*M2, NP2*( 2*NP2 +
$ M2 + MAX( NP2, N ) ) ) ) ) )
MINWRK = N*M + NP*( N + M ) + M2*M2 + NP2*NP2 +
$ MAX( 1, LW1, LW2, LW3, LW4, LW5, LW6 )
IF( LDWORK.LT.MINWRK )
$ INFO = -27
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SB10FD', -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( 1 ) = ONE
RCOND( 2 ) = ONE
RCOND( 3 ) = ONE
RCOND( 4 ) = ONE
DWORK( 1 ) = ONE
RETURN
END IF
C
TOLL = TOL
IF( TOLL.LE.ZERO ) THEN
C
C Set the default value of the tolerance.
C
TOLL = SQRT( DLAMCH( 'Epsilon' ) )
END IF
C
C Workspace usage.
C
IWC = 1 + N*M
IWD = IWC + NP*N
IWTU = IWD + NP*M
IWTY = IWTU + M2*M2
IWRK = IWTY + NP2*NP2
C
CALL DLACPY( 'Full', N, M, B, LDB, DWORK, N )
CALL DLACPY( 'Full', NP, N, C, LDC, DWORK( IWC ), NP )
CALL DLACPY( 'Full', NP, M, D, LDD, DWORK( IWD ), NP )
C
C Transform the system so that D12 and D21 satisfy the formulas
C in the computation of the Hinf (sub)optimal controller.
C
CALL SB10PD( N, M, NP, NCON, NMEAS, A, LDA, DWORK, N,
$ DWORK( IWC ), NP, DWORK( IWD ), NP, DWORK( IWTU ),
$ M2, DWORK( IWTY ), NP2, RCOND, TOLL, DWORK( IWRK ),
$ LDWORK-IWRK+1, INFO2 )
IF( INFO2.GT.0 ) THEN
INFO = INFO2
RETURN
END IF
LWAMAX = INT( DWORK( IWRK ) ) + IWRK - 1
C
IWX = IWRK
IWY = IWX + N*N
IWF = IWY + N*N
IWH = IWF + M*N
IWRK = IWH + N*NP
C
C Compute the (sub)optimal state feedback and output injection
C matrices.
C
CALL SB10QD( N, M, NP, NCON, NMEAS, GAMMA, A, LDA, DWORK, N,
$ DWORK( IWC ), NP, DWORK( IWD ), NP, DWORK( IWF ),
$ M, DWORK( IWH ), N, DWORK( IWX ), N, DWORK( IWY ),
$ N, RCOND(3), IWORK, DWORK( IWRK ), LDWORK-IWRK+1,
$ BWORK, INFO2 )
IF( INFO2.GT.0 ) THEN
INFO = INFO2 + 5
RETURN
END IF
LWAMAX = MAX( INT( DWORK( IWRK ) ) + IWRK - 1, LWAMAX )
C
C Compute the Hinf (sub)optimal controller.
C
CALL SB10RD( N, M, NP, NCON, NMEAS, GAMMA, A, LDA, DWORK, N,
$ DWORK( IWC ), NP, DWORK( IWD ), NP, DWORK( IWF ),
$ M, DWORK( IWH ), N, DWORK( IWTU ), M2, DWORK( IWTY ),
$ NP2, DWORK( IWX ), N, DWORK( IWY ), N, AK, LDAK, BK,
$ LDBK, CK, LDCK, DK, LDDK, IWORK, DWORK( IWRK ),
$ LDWORK-IWRK+1, INFO2 )
IF( INFO2.EQ.1 ) THEN
INFO = 6
RETURN
ELSE IF( INFO2.EQ.2 ) THEN
INFO = 9
RETURN
END IF
LWAMAX = MAX( INT( DWORK( IWRK ) ) + IWRK - 1, LWAMAX )
C
DWORK( 1 ) = DBLE( LWAMAX )
RETURN
C *** Last line of SB10FD ***
END