## TB01KD

### Similarity reduction of system state-matrix to block-diagonal form with two blocks so that the leading block has eigenvalues in a specified domain

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

Purpose

      To compute an additive spectral decomposition of the transfer-
      function matrix of the system (A,B,C) by reducing the system
      state-matrix A to a block-diagonal form.
      The system matrices are transformed as
      A <-- inv(U)*A*U, B <--inv(U)*B and C <-- C*U.
      The leading diagonal block of the resulting A has eigenvalues
      in a suitably defined domain of interest.

Specification

          SUBROUTINE TB01KD( DICO, STDOM, JOBA, N, M, P, ALPHA, A, LDA, B,
         $                   LDB, C, LDC, NDIM, U, LDU, WR, WI, DWORK,
         $                   LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER        DICO, JOBA, STDOM
          INTEGER          INFO, LDA, LDB, LDC, LDU, LDWORK, M, N, NDIM, P
          DOUBLE PRECISION ALPHA
    C     .. Array Arguments ..
          DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), U(LDU,*),
         $                 WI(*), WR(*)

Arguments

Mode Parameters

      DICO    CHARACTER*1
              Specifies the type of the system as follows:
              = 'C':  continuous-time system;
              = 'D':  discrete-time system.

      STDOM   CHARACTER*1
              Specifies whether the domain of interest is of stability
              type (left part of complex plane or inside of a circle)
              or of instability type (right part of complex plane or
              outside of a circle) as follows:
              = 'S':  stability type domain;
              = 'U':  instability type domain.

      JOBA    CHARACTER*1
              Specifies the shape of the state dynamics matrix on entry
              as follows:
              = 'S':  A is in an upper real Schur form;
              = 'G':  A is a general square dense matrix.

Input/Output Parameters

      N       (input) INTEGER
              The order of the state-space representation,
              i.e. the order of the matrix A.  N >= 0.

      M       (input) INTEGER
              The number of system inputs, or of columns of B.  M >= 0.

      P       (input) INTEGER
              The number of system outputs, or of rows of C.  P >= 0.

      ALPHA   (input) DOUBLE PRECISION.
              Specifies the boundary of the domain of interest for the
              eigenvalues of A. For a continuous-time system
              (DICO = 'C'), ALPHA is the boundary value for the real
              parts of eigenvalues, while for a discrete-time system
              (DICO = 'D'), ALPHA >= 0 represents the boundary value for
              the moduli of eigenvalues.

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading N-by-N part of this array must
              contain the unreduced state dynamics matrix A.
              If JOBA = 'S' then A must be a matrix in real Schur form.
              On exit, the leading N-by-N part of this array contains a
              block diagonal matrix inv(U) * A * U with two diagonal
              blocks in real Schur form with the elements below the
              first subdiagonal set to zero.
              The leading NDIM-by-NDIM block of A has eigenvalues in the
              domain of interest and the trailing (N-NDIM)-by-(N-NDIM)
              block has eigenvalues outside the domain of interest.
              The domain of interest for lambda(A), the eigenvalues
              of A, is defined by the parameters ALPHA, DICO and STDOM
              as follows:
              For a continuous-time system (DICO = 'C'):
                Real(lambda(A)) < ALPHA if STDOM = 'S';
                Real(lambda(A)) > ALPHA if STDOM = 'U';
              For a discrete-time system (DICO = 'D'):
                Abs(lambda(A)) < ALPHA if STDOM = 'S';
                Abs(lambda(A)) > ALPHA if STDOM = 'U'.

      LDA     INTEGER
              The leading dimension of array A.  LDA >= MAX(1,N).

      B       (input/output) DOUBLE PRECISION array, dimension (LDB,M)
              On entry, the leading N-by-M part of this array must
              contain the input matrix B.
              On exit, the leading N-by-M part of this array contains
              the transformed input matrix inv(U) * B.

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

      C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
              On entry, the leading P-by-N part of this array must
              contain the output matrix C.
              On exit, the leading P-by-N part of this array contains
              the transformed output matrix C * U.

      LDC     INTEGER
              The leading dimension of array C.  LDC >= MAX(1,P).

      NDIM    (output) INTEGER
              The number of eigenvalues of A lying inside the domain of
              interest for eigenvalues.

      U       (output) DOUBLE PRECISION array, dimension (LDU,N)
              The leading N-by-N part of this array contains the
              transformation matrix used to reduce A to the block-
              diagonal form. The first NDIM columns of U span the
              invariant subspace of A corresponding to the eigenvalues
              of its leading diagonal block. The last N-NDIM columns
              of U span the reducing subspace of A corresponding to
              the eigenvalues of the trailing diagonal block of A.

      LDU     INTEGER
              The leading dimension of array U.  LDU >= max(1,N).

      WR, WI  (output) DOUBLE PRECISION arrays, dimension (N)
              WR and WI contain the real and imaginary parts,
              respectively, of the computed eigenvalues of A. The
              eigenvalues will be in the same order that they appear on
              the diagonal of the output real Schur form of A. Complex
              conjugate pairs of eigenvalues will appear consecutively
              with the eigenvalue having the positive imaginary part
              first.

Workspace

      DWORK   DOUBLE PRECISION array, dimension (LDWORK)
              On exit, if INFO = 0, DWORK(1) returns the optimal value
              of LDWORK.

      LDWORK  INTEGER
              The dimension of working array DWORK.
              LDWORK >= MAX(1,N)   if JOBA = 'S';
              LDWORK >= MAX(1,3*N) if JOBA = 'G'.
              For optimum performance LDWORK should be larger.

Error Indicator

      INFO    INTEGER
              = 0: successful exit;
              < 0: if INFO = -i, the i-th argument had an illegal
                   value;
              = 1: the QR algorithm failed to compute all the
                   eigenvalues of A;
              = 2: a failure occured during the ordering of the real
                   Schur form of A;
              = 3: the separation of the two diagonal blocks failed
                   because of very close eigenvalues.

Method

      A similarity transformation U is determined that reduces the
      system state-matrix A to a block-diagonal form (with two diagonal
      blocks), so that the leading diagonal block of the resulting A has
      eigenvalues in a specified domain of the complex plane. The
      determined transformation is applied to the system (A,B,C) as
        A <-- inv(U)*A*U, B <-- inv(U)*B and C <-- C*U.

References

      [1] Safonov, M.G., Jonckheere, E.A., Verma, M., Limebeer, D.J.N.
          Synthesis of positive real multivariable feedback systems.
          Int. J. Control, pp. 817-842, 1987.

Numerical Aspects

                                      3
      The algorithm requires about 14N  floating point operations.

Further Comments

      None

Example

Program Text

    *     TB01KD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX, MMAX, PMAX
          PARAMETER        ( NMAX = 20, MMAX = 20, PMAX = 20 )
          INTEGER          LDA, LDB, LDC, LDU
          PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
         $                   LDU = NMAX )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = 3*NMAX )
    *     .. Local Scalars ..
          CHARACTER*1      DICO, JOBA, STDOM
          INTEGER          I, INFO, J, M, N, NDIM, P
          DOUBLE PRECISION ALPHA
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
         $                 DWORK(LDWORK), U(LDU,NMAX), WI(NMAX), WR(NMAX)
    *     .. External Subroutines ..
          EXTERNAL         TB01KD
    *     .. 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, ALPHA, DICO, STDOM, JOBA
          IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99990 ) 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 = 99989 ) 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 = 99988 ) P
                ELSE
                   READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
    *              Find the transformed ssr for (A,B,C).
                   CALL TB01KD( DICO, STDOM, JOBA, N, M, P, ALPHA, A, LDA,
         $                      B, LDB, C, LDC, NDIM, U, LDU, WR, WI, DWORK,
         $                      LDWORK, INFO )
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      WRITE ( NOUT, FMT = 99987 ) NDIM
                      WRITE ( NOUT, FMT = 99997 )
                      DO 10 I = 1, N
                         WRITE ( NOUT, FMT = 99994 ) WR(I), WI(I)
       10             CONTINUE
                      WRITE ( NOUT, FMT = 99996 )
                      DO 20 I = 1, N
                         WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
       20             CONTINUE
                      WRITE ( NOUT, FMT = 99993 )
                      DO 40 I = 1, N
                         WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
       40             CONTINUE
                      WRITE ( NOUT, FMT = 99992 )
                      DO 60 I = 1, P
                         WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
       60             CONTINUE
                      WRITE ( NOUT, FMT = 99991 )
                      DO 70 I = 1, N
                         WRITE ( NOUT, FMT = 99995 ) ( U(I,J), J = 1,N )
       70             CONTINUE
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' TB01KD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from TB01KD = ',I2)
    99997 FORMAT (/' The eigenvalues of state dynamics matrix A are ')
    99996 FORMAT (/' The transformed state dynamics matrix inv(U)*A*U is ')
    99995 FORMAT (20(1X,F8.4))
    99994 FORMAT ( ' (',F8.4,', ',F8.4,' )')
    99993 FORMAT (/' The transformed input/state matrix inv(U)*B is ')
    99992 FORMAT (/' The transformed state/output matrix C*U is ')
    99991 FORMAT (/' The similarity transformation matrix U is ')
    99990 FORMAT (/' N is out of range.',/' N = ',I5)
    99989 FORMAT (/' M is out of range.',/' M = ',I5)
    99988 FORMAT (/' P is out of range.',/' P = ',I5)
    99987 FORMAT (' The number of eigenvalues in the domain of interest =',
         $        I5 )
          END

Program Data

     TB01KD EXAMPLE PROGRAM DATA (Continuous system)
      5     2     3    -1.0      C     U     G
      -0.04165    4.9200   -4.9200         0         0
     -1.387944   -3.3300         0         0         0
        0.5450         0         0   -0.5450         0
             0         0    4.9200  -0.04165    4.9200
             0         0         0 -1.387944   -3.3300
             0         0
        3.3300         0
             0         0
             0         0
             0    3.3300
         1     0     0     0     0
         0     0     1     0     0
         0     0     0     1     0

Program Results

     TB01KD EXAMPLE PROGRAM RESULTS

     The number of eigenvalues in the domain of interest =    2

     The eigenvalues of state dynamics matrix A are
     ( -0.7483,   2.9940 )
     ( -0.7483,  -2.9940 )
     ( -1.6858,   2.0311 )
     ( -1.6858,  -2.0311 )
     ( -1.8751,   0.0000 )

     The transformed state dynamics matrix inv(U)*A*U is
      -0.7483  -8.6406   0.0000   0.0000   0.0000
       1.0374  -0.7483   0.0000   0.0000   0.0000
       0.0000   0.0000  -1.6858   5.5669   0.0000
       0.0000   0.0000  -0.7411  -1.6858   0.0000
       0.0000   0.0000   0.0000   0.0000  -1.8751

     The transformed input/state matrix inv(U)*B is
       2.0240  -2.0240
      -1.1309   1.1309
      -0.8621  -0.8621
       2.1912   2.1912
      -1.5555   1.5555

     The transformed state/output matrix C*U is
       0.6864  -0.0987   0.6580   0.2589   0.9650
      -0.0471   0.6873   0.0000   0.0000  -0.5609
      -0.6864   0.0987   0.6580   0.2589  -0.9650

     The similarity transformation matrix U is
       0.6864  -0.0987   0.6580   0.2589   0.9650
      -0.1665  -0.5041  -0.2589   0.6580  -0.9205
      -0.0471   0.6873   0.0000   0.0000  -0.5609
      -0.6864   0.0987   0.6580   0.2589  -0.9650
       0.1665   0.5041  -0.2589   0.6580   0.9205

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