SUBROUTINE SB10QD( N, M, NP, NCON, NMEAS, GAMMA, A, LDA, B, LDB,
$ C, LDC, D, LDD, 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 H-infinity (sub)optimal n-state controller,
C using 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 with D12 = | 0 | and D21 is
C | I |
C full row rank with D21 = | 0 I | as obtained by the
C subroutine SB10PD,
C
C (A3) | A-j*omega*I B2 | has full column rank for all omega,
C | C1 D12 |
C
C
C (A4) | A-j*omega*I B1 | has full row rank for all omega.
C | C2 D21 |
C
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 F (output) DOUBLE PRECISION array, dimension (LDF,N)
C The leading M-by-N part of this array contains the state
C feedback matrix F.
C
C LDF INTEGER
C The leading dimension of the array F. LDF >= max(1,M).
C
C H (output) DOUBLE PRECISION array, dimension (LDH,NP)
C The leading N-by-NP part of this array contains the output
C 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
C (max(2*max(N,M-NCON,NP-NMEAS),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 >= max(1,M*M + max(2*M1,3*N*N +
C 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 where M1 = M - M2 and NP1 = NP - NP2.
C For good performance, LDWORK must generally be larger.
C Denoting Q = MAX(M1,M2,NP1,NP2), an upper bound is
C max(1,4*Q*Q+max(2*Q,3*N*N + max(2*N*Q,10*N*N+12*N+5))).
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 controller is not admissible (too small value
C of gamma);
C = 2: if the X-Riccati equation was not solved
C successfully (the controller is not admissible or
C there are numerical difficulties);
C = 3: if the Y-Riccati equation was not solved
C successfully (the controller is not admissible or
C there are numerical difficulties).
C
C METHOD
C
C The routine implements the Glover's and Doyle's formulas [1],[2]
C modified as described in [3]. The X- and Y-Riccati equations
C are solved with condition and accuracy estimates [4].
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 [4] 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 Sept. 1999.
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, LDB, LDC, LDD, LDF, LDH, LDWORK,
$ LDX, LDY, M, N, NCON, NMEAS, NP
DOUBLE PRECISION GAMMA
C ..
C .. Array Arguments ..
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ D( LDD, * ), DWORK( * ), F( LDF, * ),
$ H( LDH, * ), X( LDX, * ), XYCOND( 2 ),
$ Y( LDY, * )
LOGICAL BWORK( * )
C
C ..
C .. Local Scalars ..
INTEGER INFO2, IW2, IWA, IWG, IWI, IWQ, IWR, IWRK, IWS,
$ IWT, IWV, LWAMAX, M1, M2, MINWRK, N2, ND1, ND2,
$ NN, NP1, NP2
DOUBLE PRECISION ANORM, EPS, FERR, RCOND, SEP
C ..
C .. External Functions ..
C
DOUBLE PRECISION DLAMCH, DLANSY
EXTERNAL DLAMCH, DLANSY
C ..
C .. External Subroutines ..
EXTERNAL DGEMM, DLACPY, DLASET, DSYCON, DSYMM, DSYRK,
$ DSYTRF, DSYTRI, MB01RU, MB01RX, 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
NN = N*N
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( LDF.LT.MAX( 1, M ) ) THEN
INFO = -16
ELSE IF( LDH.LT.MAX( 1, N ) ) THEN
INFO = -18
ELSE IF( LDX.LT.MAX( 1, N ) ) THEN
INFO = -20
ELSE IF( LDY.LT.MAX( 1, N ) ) THEN
INFO = -22
ELSE
C
C Compute workspace.
C
MINWRK = MAX( 1, M*M + MAX( 2*M1, 3*NN +
$ MAX( N*M, 10*NN + 12*N + 5 ) ),
$ NP*NP + MAX( 2*NP1, 3*NN +
$ MAX( N*NP, 10*NN + 12*N + 5 ) ) )
IF( LDWORK.LT.MINWRK )
$ INFO = -26
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SB10QD', -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
XYCOND( 1 ) = ONE
XYCOND( 2 ) = ONE
DWORK( 1 ) = ONE
RETURN
END IF
ND1 = NP1 - M2
ND2 = M1 - NP2
N2 = 2*N
C
C Get the machine precision.
C
EPS = DLAMCH( 'Epsilon' )
C
C Workspace usage.
C
IWA = M*M + 1
IWQ = IWA + NN
IWG = IWQ + NN
IW2 = IWG + NN
C
C Compute |D1111'||D1111 D1112| - gamma^2*Im1 .
C |D1112'|
C
CALL DLASET( 'L', M1, M1, ZERO, -GAMMA*GAMMA, DWORK, M )
IF( ND1.GT.0 )
$ CALL DSYRK( 'L', 'T', M1, ND1, ONE, D, LDD, ONE, DWORK, M )
C
C Compute inv(|D1111'|*|D1111 D1112| - gamma^2*Im1) .
C |D1112'|
C
IWRK = IWA
ANORM = DLANSY( 'I', 'L', M1, DWORK, M, DWORK( IWRK ) )
CALL DSYTRF( 'L', M1, DWORK, M, IWORK, DWORK( IWRK ),
$ LDWORK-IWRK+1, INFO2 )
IF( INFO2.GT.0 ) THEN
INFO = 1
RETURN
END IF
C
LWAMAX = INT( DWORK( IWRK ) ) + IWRK - 1
CALL DSYCON( 'L', M1, DWORK, M, IWORK, ANORM, RCOND,
$ DWORK( IWRK ), IWORK( M1+1 ), INFO2 )
IF( RCOND.LT.EPS ) THEN
INFO = 1
RETURN
END IF
C
C Compute inv(R) block by block.
C
CALL DSYTRI( 'L', M1, DWORK, M, IWORK, DWORK( IWRK ), INFO2 )
C
C Compute -|D1121 D1122|*inv(|D1111'|*|D1111 D1112| - gamma^2*Im1) .
C |D1112'|
C
CALL DSYMM( 'R', 'L', M2, M1, -ONE, DWORK, M, D( ND1+1, 1 ), LDD,
$ ZERO, DWORK( M1+1 ), M )
C
C Compute |D1121 D1122|*inv(|D1111'|*|D1111 D1112| -
C |D1112'|
C
C gamma^2*Im1)*|D1121'| + Im2 .
C |D1122'|
C
CALL DLASET( 'Lower', M2, M2, ZERO, ONE, DWORK( M1*(M+1)+1 ), M )
CALL MB01RX( 'Right', 'Lower', 'Transpose', M2, M1, ONE, -ONE,
$ DWORK( M1*(M+1)+1 ), M, D( ND1+1, 1 ), LDD,
$ DWORK( M1+1 ), M, INFO2 )
C
C Compute D11'*C1 .
C
CALL DGEMM( 'T', 'N', M1, N, NP1, ONE, D, LDD, C, LDC, ZERO,
$ DWORK( IW2 ), M )
C
C Compute D1D'*C1 .
C
CALL DLACPY( 'Full', M2, N, C( ND1+1, 1 ), LDC, DWORK( IW2+M1 ),
$ M )
C
C Compute inv(R)*D1D'*C1 in F .
C
CALL DSYMM( 'L', 'L', M, N, ONE, DWORK, M, DWORK( IW2 ), M, ZERO,
$ F, LDF )
C
C Compute Ax = A - B*inv(R)*D1D'*C1 .
C
CALL DLACPY( 'Full', N, N, A, LDA, DWORK( IWA ), N )
CALL DGEMM( 'N', 'N', N, N, M, -ONE, B, LDB, F, LDF, ONE,
$ DWORK( IWA ), N )
C
C Compute Cx = C1'*C1 - C1'*D1D*inv(R)*D1D'*C1 .
C
IF( ND1.EQ.0 ) THEN
CALL DLASET( 'L', N, N, ZERO, ZERO, DWORK( IWQ ), N )
ELSE
CALL DSYRK( 'L', 'T', N, NP1, ONE, C, LDC, ZERO,
$ DWORK( IWQ ), N )
CALL MB01RX( 'Left', 'Lower', 'Transpose', N, M, ONE, -ONE,
$ DWORK( IWQ ), N, DWORK( IW2 ), M, F, LDF, INFO2 )
END IF
C
C Compute Dx = B*inv(R)*B' .
C
IWRK = IW2
CALL MB01RU( 'Lower', 'NoTranspose', N, M, ZERO, ONE,
$ DWORK( IWG ), N, B, LDB, DWORK, M, DWORK( IWRK ),
$ M*N, INFO2 )
C
C Solution of the Riccati equation Ax'*X + X*Ax + Cx - X*Dx*X = 0 .
C Workspace: need M*M + 13*N*N + 12*N + 5;
C prefer larger.
C
IWT = IW2
IWV = IWT + NN
IWR = IWV + NN
IWI = IWR + N2
IWS = IWI + N2
IWRK = IWS + 4*NN
C
CALL SB02RD( 'All', 'Continuous', 'NotUsed', 'NoTranspose',
$ 'Lower', 'GeneralScaling', 'Stable', 'NotFactored',
$ 'Original', N, DWORK( IWA ), 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 = 2
RETURN
END IF
C
LWAMAX = MAX( INT( DWORK( IWRK ) ) + IWRK - 1, LWAMAX )
C
C Compute F = -inv(R)*|D1D'*C1 + B'*X| .
C
IWRK = IW2
CALL DGEMM( 'T', 'N', M, N, N, ONE, B, LDB, X, LDX, ZERO,
$ DWORK( IWRK ), M )
CALL DSYMM( 'L', 'L', M, N, -ONE, DWORK, M, DWORK( IWRK ), M,
$ -ONE, F, LDF )
C
C Workspace usage.
C
IWA = NP*NP + 1
IWQ = IWA + NN
IWG = IWQ + NN
IW2 = IWG + NN
C
C Compute |D1111|*|D1111' D1121'| - gamma^2*Inp1 .
C |D1121|
C
CALL DLASET( 'U', NP1, NP1, ZERO, -GAMMA*GAMMA, DWORK, NP )
IF( ND2.GT.0 )
$ CALL DSYRK( 'U', 'N', NP1, ND2, ONE, D, LDD, ONE, DWORK, NP )
C
C Compute inv(|D1111|*|D1111' D1121'| - gamma^2*Inp1) .
C |D1121|
C
IWRK = IWA
ANORM = DLANSY( 'I', 'U', NP1, DWORK, NP, DWORK( IWRK ) )
CALL DSYTRF( 'U', NP1, DWORK, NP, IWORK, DWORK( IWRK ),
$ LDWORK-IWRK+1, INFO2 )
IF( INFO2.GT.0 ) THEN
INFO = 1
RETURN
END IF
C
LWAMAX = MAX( INT( DWORK( IWRK ) ) + IWRK - 1, LWAMAX )
CALL DSYCON( 'U', NP1, DWORK, NP, IWORK, ANORM, RCOND,
$ DWORK( IWRK ), IWORK( NP1+1 ), INFO2 )
IF( RCOND.LT.EPS ) THEN
INFO = 1
RETURN
END IF
C
C Compute inv(RT) .
C
CALL DSYTRI( 'U', NP1, DWORK, NP, IWORK, DWORK( IWRK ), INFO2 )
C
C Compute -inv(|D1111||D1111' D1121'| - gamma^2*Inp1)*|D1112| .
C |D1121| |D1122|
C
CALL DSYMM( 'L', 'U', NP1, NP2, -ONE, DWORK, NP, D( 1, ND2+1 ),
$ LDD, ZERO, DWORK( NP1*NP+1 ), NP )
C
C Compute [D1112' D1122']*inv(|D1111||D1111' D1121'| -
C |D1121|
C
C gamma^2*Inp1)*|D1112| + Inp2 .
C |D1122|
C
CALL DLASET( 'Full', NP2, NP2, ZERO, ONE, DWORK( NP1*(NP+1)+1 ),
$ NP )
CALL MB01RX( 'Left', 'Upper', 'Transpose', NP2, NP1, ONE, -ONE,
$ DWORK( NP1*(NP+1)+1 ), NP, D( 1, ND2+1 ), LDD,
$ DWORK( NP1*NP+1 ), NP, INFO2 )
C
C Compute B1*D11' .
C
CALL DGEMM( 'N', 'T', N, NP1, M1, ONE, B, LDB, D, LDD, ZERO,
$ DWORK( IW2 ), N )
C
C Compute B1*DD1' .
C
CALL DLACPY( 'Full', N, NP2, B( 1, ND2+1 ), LDB,
$ DWORK( IW2+NP1*N ), N )
C
C Compute B1*DD1'*inv(RT) in H .
C
CALL DSYMM( 'R', 'U', N, NP, ONE, DWORK, NP, DWORK( IW2 ), N,
$ ZERO, H, LDH )
C
C Compute Ay = A - B1*DD1'*inv(RT)*C .
C
CALL DLACPY( 'Full', N, N, A, LDA, DWORK( IWA ), N )
CALL DGEMM( 'N', 'N', N, N, NP, -ONE, H, LDH, C, LDC, ONE,
$ DWORK( IWA ), N )
C
C Compute Cy = B1*B1' - B1*DD1'*inv(RT)*DD1*B1' .
C
IF( ND2.EQ.0 ) THEN
CALL DLASET( 'U', N, N, ZERO, ZERO, DWORK( IWQ ), N )
ELSE
CALL DSYRK( 'U', 'N', N, M1, ONE, B, LDB, ZERO, DWORK( IWQ ),
$ N )
CALL MB01RX( 'Right', 'Upper', 'Transpose', N, NP, ONE, -ONE,
$ DWORK( IWQ ), N, H, LDH, DWORK( IW2 ), N, INFO2 )
END IF
C
C Compute Dy = C'*inv(RT)*C .
C
IWRK = IW2
CALL MB01RU( 'Upper', 'Transpose', N, NP, ZERO, ONE, DWORK( IWG ),
$ N, C, LDC, DWORK, NP, DWORK( IWRK), N*NP, INFO2 )
C
C Solution of the Riccati equation Ay*Y + Y*Ay' + Cy - Y*Dy*Y = 0 .
C Workspace: need NP*NP + 13*N*N + 12*N + 5;
C prefer larger.
C
IWT = IW2
IWV = IWT + NN
IWR = IWV + NN
IWI = IWR + N2
IWS = IWI + N2
IWRK = IWS + 4*NN
C
CALL SB02RD( 'All', 'Continuous', 'NotUsed', 'Transpose',
$ 'Upper', 'GeneralScaling', 'Stable', 'NotFactored',
$ 'Original', N, DWORK( IWA ), 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 = 3
RETURN
END IF
C
LWAMAX = MAX( INT( DWORK( IWRK ) ) + IWRK - 1, LWAMAX )
C
C Compute H = -|B1*DD1' + Y*C'|*inv(RT) .
C
IWRK = IW2
CALL DGEMM( 'N', 'T', N, NP, N, ONE, Y, LDY, C, LDC, ZERO,
$ DWORK( IWRK ), N )
CALL DSYMM( 'R', 'U', N, NP, -ONE, DWORK, NP, DWORK( IWRK ), N,
$ -ONE, H, LDH )
C
DWORK( 1 ) = DBLE( LWAMAX )
RETURN
C *** Last line of SB10QD ***
END