## SB02ND

### Optimal state feedback matrix for an optimal control problem

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

      To compute the optimal feedback matrix F for the problem of
      optimal control given by

                         -1
           F = (R + B'XB)  (B'XA + L')                           (1)

      in the discrete-time case and

                -1
           F = R  (B'X + L')                                     (2)

      in the continuous-time case, where A, B and L are N-by-N, N-by-M
      and N-by-M matrices respectively; R and X are M-by-M and N-by-N
      symmetric matrices respectively.

      Optionally, matrix R may be specified in a factored form, and L
      may be zero.

Specification

          SUBROUTINE SB02ND( DICO, FACT, UPLO, JOBL, N, M, P, A, LDA, B,
         $                   LDB, R, LDR, IPIV, L, LDL, X, LDX, RNORM, F,
         $                   LDF, OUFACT, IWORK, DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         DICO, FACT, JOBL, UPLO
          INTEGER           INFO, LDA, LDB, LDF, LDL, LDR, LDWORK, LDX, M,
         $                  N, P
          DOUBLE PRECISION  RNORM
    C     .. Array Arguments ..
          INTEGER           IPIV(*), IWORK(*), OUFACT(2)
          DOUBLE PRECISION  A(LDA,*), B(LDB,*), DWORK(*), F(LDF,*),
         $                  L(LDL,*), R(LDR,*), X(LDX,*)

Arguments

Mode Parameters

      DICO    CHARACTER*1
              Specifies the equation from which F is to be determined,
              as follows:
              = 'D':  Equation (1), discrete-time case;
              = 'C':  Equation (2), continuous-time case.

      FACT    CHARACTER*1
              Specifies how the matrix R is given (factored or not), as
              follows:
              = 'N':  Array R contains the matrix R;
              = 'D':  Array R contains a P-by-M matrix D, where R = D'D;
              = 'C':  Array R contains the Cholesky factor of R;
              = 'U':  Array R contains the symmetric indefinite UdU' or
                      LdL' factorization of R. This option is not
                      available for DICO = 'D'.

      UPLO    CHARACTER*1
              Specifies which triangle of the possibly factored matrix R
              (or R + B'XB, on exit) is or should be stored, as follows:
              = 'U':  Upper triangle is stored;
              = 'L':  Lower triangle is stored.

      JOBL    CHARACTER*1
              Specifies whether or not the matrix L is zero, as follows:
              = 'Z':  L is zero;
              = 'N':  L is nonzero.

Input/Output Parameters

      N       (input) INTEGER
              The order of the matrices A and X.  N >= 0.
              No computations are performed if MIN(N,M) = 0.

      M       (input) INTEGER
              The number of system inputs.  M >= 0.

      P       (input) INTEGER
              The number of rows of the matrix D.
              P >= M for DICO = 'C';
              P >= 0 for DICO = 'D'.
              This parameter must be specified only for FACT = 'D'.

      A       (input) DOUBLE PRECISION array, dimension (LDA,N)
              If DICO = 'D', the leading N-by-N part of this array must
              contain the state matrix A of the system.
              If DICO = 'C', this array is not referenced.

      LDA     INTEGER
              The leading dimension of array A.
              LDA >= MAX(1,N) if DICO = 'D';
              LDA >= 1        if DICO = 'C'.

      B       (input/worksp.) DOUBLE PRECISION array, dimension (LDB,M)
              The leading N-by-M part of this array must contain the
              input matrix B of the system.
              If DICO = 'D' and FACT = 'D' or 'C', the contents of this
              array is destroyed. Specifically, if, on exit,
              OUFACT(2) = 1, this array contains chol(X)*B, and if
              OUFACT(2) = 2 and INFO < M+2, but INFO >= 0, its trailing
              part (in the first N rows) contains the submatrix of
              sqrt(V)*U'B corresponding to the non-negligible, positive
              eigenvalues of X, where V and U are the matrices with the
              eigenvalues and eigenvectors of X.
              Otherwise, B is unchanged on exit.

      LDB     INTEGER
              The leading dimension of array B.  LDB >= MAX(1,N).

      R       (input/output) DOUBLE PRECISION array, dimension (LDR,M)
              On entry, if FACT = 'N', the leading M-by-M upper
              triangular part (if UPLO = 'U') or lower triangular part
              (if UPLO = 'L') of this array must contain the upper
              triangular part or lower triangular part, respectively,
              of the symmetric input weighting matrix R.
              On entry, if FACT = 'D', the leading P-by-M part of this
              array must contain the direct transmission matrix D of the
              system.
              On entry, if FACT = 'C', the leading M-by-M upper
              triangular part (if UPLO = 'U') or lower triangular part
              (if UPLO = 'L') of this array must contain the Cholesky
              factor of the positive definite input weighting matrix R
              (as produced by LAPACK routine DPOTRF).
              On entry, if DICO = 'C' and FACT = 'U', the leading M-by-M
              upper triangular part (if UPLO = 'U') or lower triangular
              part (if UPLO = 'L') of this array must contain the
              factors of the UdU' or LdL' factorization, respectively,
              of the symmetric indefinite input weighting matrix R (as
              produced by LAPACK routine DSYTRF).
              The strictly lower triangular part (if UPLO = 'U') or
              strictly upper triangular part (if UPLO = 'L') of this
              array is used as workspace (filled in by symmetry with the
              other strictly triangular part of R, of R+B'XB, or of the
              result, if DICO = 'C', DICO = 'D', or (DICO = 'D' and
              (FACT = 'D' or FACT = 'C') and UPLO = 'L'), respectively.
              On exit, if OUFACT(1) = 1, and INFO = 0 (or INFO = M+1),
              the leading M-by-M upper triangular part (if UPLO = 'U')
              or lower triangular part (if UPLO = 'L') of this array
              contains the Cholesky factor of the given input weighting
              matrix R (for DICO = 'C'), or that of the matrix R + B'XB
              (for DICO = 'D').
              On exit, if OUFACT(1) = 2, and INFO = 0 (or INFO = M+1),
              the leading M-by-M upper triangular part (if UPLO = 'U')
              or lower triangular part (if UPLO = 'L') of this array
              contains the factors of the UdU' or LdL' factorization,
              respectively, of the given input weighting matrix
              (for DICO = 'C'), or that of the matrix R + B'XB
              (for DICO = 'D' and FACT = 'N').
              On exit R is unchanged if FACT = 'U' or N = 0.

      LDR     INTEGER.
              The leading dimension of the array R.
              LDR >= MAX(1,M)   if FACT <> 'D';
              LDR >= MAX(1,M,P) if FACT =  'D'.

      IPIV    (input/output) INTEGER array, dimension (M)
              On entry, if FACT = 'U', this array must contain details
              of the interchanges performed and the block structure of
              the d factor in the UdU' or LdL' factorization of matrix R
              (as produced by LAPACK routine DSYTRF).
              On exit, if OUFACT(1) = 2, this array contains details of
              the interchanges performed and the block structure of the
              d factor in the UdU' or LdL' factorization of matrix R or
              R + B'XB, as produced by LAPACK routine DSYTRF.
              This array is not referenced if FACT = 'D', or FACT = 'C',
              or N = 0.

      L       (input) DOUBLE PRECISION array, dimension (LDL,M)
              If JOBL = 'N', the leading N-by-M part of this array must
              contain the cross weighting matrix L.
              If JOBL = 'Z', this array is not referenced.

      LDL     INTEGER
              The leading dimension of array L.
              LDL >= MAX(1,N) if JOBL = 'N';
              LDL >= 1        if JOBL = 'Z'.

      X       (input/output) DOUBLE PRECISION array, dimension (LDX,N)
              On entry, the leading N-by-N part of this array must
              contain the solution matrix X of the algebraic Riccati
              equation as produced by SLICOT Library routines SB02MD or
              SB02OD. Matrix X is assumed non-negative definite if
              DICO = 'D' and (FACT = 'D' or FACT = 'C').
              The full matrix X must be given on input if LDWORK < N*M
              or if DICO = 'D' and (FACT = 'D' or FACT = 'C').
              On exit, if DICO = 'D', FACT = 'D' or FACT = 'C', and
              OUFACT(2) = 1, the N-by-N upper triangular part
              (if UPLO = 'U') or lower triangular part (if UPLO = 'L')
              of this array contains the Cholesky factor of the given
              matrix X, which is found to be positive definite.
              On exit, if DICO = 'D', FACT = 'D' or 'C', OUFACT(2) = 2,
              and INFO < M+2 (but INFO >= 0), the leading N-by-N part of
              this array contains the matrix of orthonormal eigenvectors
              of X.
              On exit X is unchanged if DICO = 'C' or FACT = 'N'.

      LDX     INTEGER
              The leading dimension of array X.  LDX >= MAX(1,N).

      RNORM   (input) DOUBLE PRECISION
              If FACT = 'U', this parameter must contain the 1-norm of
              the original matrix R (before factoring it).
              Otherwise, this parameter is not used.

      F       (output) DOUBLE PRECISION array, dimension (LDF,N)
              The leading M-by-N part of this array contains the
              optimal feedback matrix F.
              This array is not referenced if DICO = 'C' and FACT = 'D'
              and P < M.

      LDF     INTEGER
              The leading dimension of array F.  LDF >= MAX(1,M).

      OUFACT  (output) INTEGER array, dimension (2)
              Information about the factorization finally used.
              OUFACT(1) = 1:  Cholesky factorization of R (or R + B'XB)
                              has been used;
              OUFACT(1) = 2:  UdU' (if UPLO = 'U') or LdL' (if UPLO =
                              'L') factorization of R (or R + B'XB)
                              has been used;
              OUFACT(2) = 1:  Cholesky factorization of X has been used;
              OUFACT(2) = 2:  Spectral factorization of X has been used.
              The value of OUFACT(2) is not set for DICO = 'C' or for
              DICO = 'D' and FACT = 'N'.
              This array is not set if N = 0 or M = 0.

Workspace

      IWORK   INTEGER array, dimension (M)

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0 or LDWORK = -1, DWORK(1) returns the
              optimal value of LDWORK, and for LDWORK set as specified
              below, DWORK(2) contains the reciprocal condition number
              of the matrix R (for DICO = 'C') or of R + B'XB (for
              DICO = 'D'); DWORK(2) is set to 1 if N = 0.
              On exit, if LDWORK = -2 on input or INFO = -25, then
              DWORK(1) returns the minimal value of LDWORK.
              If on exit INFO = 0, and OUFACT(2) = 2, then DWORK(3),...,
              DWORK(N+2) contain the eigenvalues of X, in ascending
              order.

      LDWORK  INTEGER
              Dimension of working array DWORK.
              LDWORK >= max(2,2*M)           if FACT =  'U';
              LDWORK >= max(2,3*M)           if FACT <> 'U', DICO = 'C';
              LDWORK >= max(2,3*M,N)         if FACT =  'N', DICO = 'D';
              LDWORK >= max(N+3*M+2,4*N+1)   if FACT <> 'N', DICO = 'D'.
              For optimum performance LDWORK should be larger.

              If LDWORK = -1, an optimal workspace query is assumed; the
              routine only calculates the optimal size of the DWORK
              array, returns this value as the first entry of the DWORK
              array, and no error message is issued by XERBLA.

              If LDWORK = -2, a minimal workspace query is assumed; the
              routine only calculates the minimal size of the DWORK
              array, returns this value as the first entry of the DWORK
              array, and no error message is issued by XERBLA.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              = i:  if the i-th element of the d factor is exactly zero;
                    the UdU' (or LdL') factorization has been completed,
                    but the block diagonal matrix d is exactly singular;
              = M+1:  if the matrix R (if DICO = 'C'), or R + B'XB
                    (if DICO = 'D') is numerically singular (to working
                    precision);
              = M+2:  if one or more of the eigenvalues of X has not
                    converged;
              = M+3:  if the matrix X is indefinite and updating the
                    triangular factorization failed.
              If INFO > M+1, call the routine again with an appropriate,
              unfactored matrix R.

Method

      The optimal feedback matrix F is obtained as the solution to the
      system of linear equations

         (R + B'XB) * F = B'XA + L'

      in the discrete-time case and

         R * F = B'X + L'

      in the continuous-time case, with R replaced by D'D if FACT = 'D'.
      If FACT = 'N', Cholesky factorization is tried first, but
      if the coefficient matrix is not positive definite, then UdU' (or
      LdL') factorization is used. If FACT <> 'N', the factored form
      of R is taken into account. The discrete-time case then involves
      updating of a triangular factorization of R (or D'D); Cholesky or
      symmetric spectral factorization of X is employed to avoid
      squaring of the condition number of the matrix. When D is given,
      its QR factorization is determined, and the triangular factor is
      used as described above.

Numerical Aspects

      The algorithm consists of numerically stable steps.
                                     3     2
      For DICO = 'C', it requires O(m  + mn ) floating point operations
                            2
      if FACT = 'N' and O(mn ) floating point operations, otherwise.
      For DICO = 'D', the operation counts are similar, but additional
         3
      O(n ) floating point operations may be needed in the worst case.
      These estimates assume that M <= N.

Further Comments

      None

Example

Program Text

    *     SB02ND EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX, MMAX, PMAX
          PARAMETER        ( NMAX = 20, MMAX = 20, PMAX = 20 )
          INTEGER          NMAX2
          PARAMETER        ( NMAX2 = 2*NMAX )
          INTEGER          LDA, LDB, LDC, LDL, LDR, LDS, LDT, LDU, LDX, LDF
          PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX, LDL = NMAX,
         $                   LDR = MAX(MMAX,PMAX), LDS = NMAX2+MMAX,
         $                   LDT = NMAX2+MMAX, LDU = NMAX2, LDX = NMAX,
         $                   LDF = MMAX )
          INTEGER          LIWORK
          PARAMETER        ( LIWORK = MAX( NMAX2,MMAX ) )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = MAX( NMAX+3*MMAX+2, 14*NMAX+23,
         $                   16*NMAX ) )
    *     .. Local Scalars ..
          DOUBLE PRECISION TOL, RCOND, RNORM
          INTEGER          I, INFO1, INFO2, J, M, N, P
          CHARACTER*1      DICO, FACT, JOBB, JOBL, SORT, UPLO
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), ALFAI(2*NMAX), ALFAR(2*NMAX),
         $                 B(LDB,MMAX), BETA(2*NMAX), C(LDC,NMAX),
         $                 DWORK(LDWORK), F(LDF,NMAX), L(LDL,MMAX),
         $                 R(LDR,MMAX), S(LDS,NMAX2+MMAX), T(LDT,NMAX2),
         $                 U(LDU,NMAX2), X(LDX,NMAX)
          INTEGER          IPIV(LIWORK), IWORK(LIWORK), OUFACT(2)
          LOGICAL          BWORK(NMAX2)
    *     .. External Functions ..
          LOGICAL          LSAME
          EXTERNAL         LSAME
    *     .. External Subroutines ..
          EXTERNAL         SB02ND, SB02OD
    *     .. Intrinsic Functions ..
          INTRINSIC        MAX
    *     .. Executable Statements ..
    *
          WRITE ( NOUT, FMT = 99999 )
    *     Skip the heading in the data file and read the data.
          READ ( NIN, FMT = '()' )
          READ ( NIN, FMT = * ) N, M, P, TOL, DICO, FACT, JOBL, UPLO
          IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99993 ) N
          ELSE
             READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
             IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
                WRITE ( NOUT, FMT = 99992 ) M
             ELSE
                READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
                IF ( P.LT.0 .OR. P.GT.PMAX ) THEN
                   WRITE ( NOUT, FMT = 99991 ) P
                ELSE
                   READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
                   IF ( LSAME( FACT, 'D' ) ) THEN
                      READ ( NIN, FMT = * ) ( ( R(I,J), J = 1,M ), I = 1,P )
                   ELSE
                      READ ( NIN, FMT = * ) ( ( R(I,J), J = 1,M ), I = 1,M )
                   END IF
    *              Find the solution matrix X.
                   JOBB = 'B'
                   SORT = 'S'
                   CALL SB02OD( DICO, JOBB, 'Both', UPLO, JOBL, SORT, N, M,
         $                      P, A, LDA, B, LDB, C, LDC, R, LDR, L, LDL,
         $                      RCOND, X, LDX, ALFAR, ALFAI, BETA, S, LDS,
         $                      T, LDT, U, LDU, TOL, IWORK, DWORK, LDWORK,
         $                      BWORK, INFO1 )
    *
                   IF ( INFO1.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO1
                   ELSE
                      WRITE ( NOUT, FMT = 99996 )
                      DO 20 I = 1, N
                         WRITE ( NOUT, FMT = 99994 ) ( X(I,J), J = 1,N )
       20             CONTINUE
    *                 Compute the optimal feedback matrix F.
                      CALL SB02ND( DICO, FACT, UPLO, JOBL, N, M, P, A, LDA,
         $                         B, LDB, R, LDR, IPIV, L, LDL, X, LDX,
         $                         RNORM, F, LDF, OUFACT, IWORK, DWORK,
         $                         LDWORK, INFO2 )
    *
                      IF ( INFO2.NE.0 ) THEN
                         WRITE ( NOUT, FMT = 99997 ) INFO2
                      ELSE
                         WRITE ( NOUT, FMT = 99995 )
                         DO 40 I = 1, M
                            WRITE ( NOUT, FMT = 99994 ) ( F(I,J), J = 1,N )
       40                CONTINUE
                      END IF
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' SB02ND EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from SB02OD = ',I2)
    99997 FORMAT (' INFO on exit from SB02ND = ',I2)
    99996 FORMAT (' The solution matrix X is ')
    99995 FORMAT (/' The optimal feedback matrix F is ')
    99994 FORMAT (20(1X,F8.4))
    99993 FORMAT (/' N is out of range.',/' N = ',I5)
    99992 FORMAT (/' M is out of range.',/' M = ',I5)
    99991 FORMAT (/' P is out of range.',/' P = ',I5)
          END

Program Data

     SB02ND EXAMPLE PROGRAM DATA
       2     1     3     0.0     D     N     Z     U
       2.0 -1.0
       1.0  0.0
       1.0
       0.0
       0.0  0.0
       0.0  0.0
       0.0  1.0
       0.0
       0.0
       0.0

Program Results

     SB02ND EXAMPLE PROGRAM RESULTS

     The solution matrix X is
       1.0000   0.0000
       0.0000   1.0000

     The optimal feedback matrix F is
       2.0000  -1.0000

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