## AB08ND

### Construction of a regular pencil for a given system such that its generalized eigenvalues are invariant zeros of the system

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

Purpose

      To construct for a linear multivariable system described by a
      state-space model (A,B,C,D) a regular pencil (A - lambda*B ) which
                                                     f          f
      has the invariant zeros of the system as generalized eigenvalues.
      The routine also computes the orders of the infinite zeros and the
      right and left Kronecker indices of the system (A,B,C,D).

Specification

          SUBROUTINE AB08ND( EQUIL, N, M, P, A, LDA, B, LDB, C, LDC, D, LDD,
         $                   NU, RANK, DINFZ, NKROR, NKROL, INFZ, KRONR,
         $                   KRONL, AF, LDAF, BF, LDBF, TOL, IWORK, DWORK,
         $                   LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         EQUIL
          INTEGER           DINFZ, INFO, LDA, LDAF, LDB, LDBF, LDC, LDD,
         $                  LDWORK, M, N, NKROL, NKROR, NU, P, RANK
          DOUBLE PRECISION  TOL
    C     .. Array Arguments ..
          INTEGER           INFZ(*), IWORK(*), KRONL(*), KRONR(*)
          DOUBLE PRECISION  A(LDA,*), AF(LDAF,*), B(LDB,*), BF(LDBF,*),
         $                  C(LDC,*), D(LDD,*), DWORK(*)

Arguments

Mode Parameters

      EQUIL   CHARACTER*1
              Specifies whether the user wishes to balance the compound
              matrix (see METHOD) as follows:
              = 'S':  Perform balancing (scaling);
              = 'N':  Do not perform balancing.

Input/Output Parameters

      N       (input) INTEGER
              The number of state variables, i.e., the order of the
              matrix A.  N >= 0.

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

      P       (input) INTEGER
              The number of system outputs.  P >= 0.

      A       (input) DOUBLE PRECISION array, dimension (LDA,N)
              The leading N-by-N part of this array must contain the
              state dynamics matrix A of the system.

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

      B       (input) DOUBLE PRECISION array, dimension (LDB,M)
              The leading N-by-M part of this array must contain the
              input/state matrix B of the system.

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

      C       (input) DOUBLE PRECISION array, dimension (LDC,N)
              The leading P-by-N part of this array must contain the
              state/output matrix C of the system.

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

      D       (input) DOUBLE PRECISION array, dimension (LDD,M)
              The leading P-by-M part of this array must contain the
              direct transmission matrix D of the system.

      LDD     INTEGER
              The leading dimension of array D.  LDD >= MAX(1,P).

      NU      (output) INTEGER
              The number of (finite) invariant zeros.

      RANK    (output) INTEGER
              The normal rank of the transfer function matrix.

      DINFZ   (output) INTEGER
              The maximum degree of infinite elementary divisors.

      NKROR   (output) INTEGER
              The number of right Kronecker indices.

      NKROL   (output) INTEGER
              The number of left Kronecker indices.

      INFZ    (output) INTEGER array, dimension (N)
              The leading DINFZ elements of INFZ contain information
              on the infinite elementary divisors as follows:
              the system has INFZ(i) infinite elementary divisors
              of degree i, where i = 1,2,...,DINFZ.

      KRONR   (output) INTEGER array, dimension (MAX(N,M)+1)
              The leading NKROR elements of this array contain the
              right Kronecker (column) indices.

      KRONL   (output) INTEGER array, dimension (MAX(N,P)+1)
              The leading NKROL elements of this array contain the
              left Kronecker (row) indices.

      AF      (output) DOUBLE PRECISION array, dimension
              (LDAF,N+MIN(P,M))
              The leading NU-by-NU part of this array contains the
              coefficient matrix A  of the reduced pencil. The remainder
                                  f
              of the leading (N+M)-by-(N+MIN(P,M)) part is used as
              internal workspace.

      LDAF    INTEGER
              The leading dimension of array AF.  LDAF >= MAX(1,N+M).

      BF      (output) DOUBLE PRECISION array, dimension (LDBF,N+M)
              The leading NU-by-NU part of this array contains the
              coefficient matrix B  of the reduced pencil. The
                                  f
              remainder of the leading (N+P)-by-(N+M) part is used as
              internal workspace.

      LDBF    INTEGER
              The leading dimension of array BF.  LDBF >= MAX(1,N+P).

Tolerances

      TOL     DOUBLE PRECISION
              A tolerance used in rank decisions to determine the
              effective rank, which is defined as the order of the
              largest leading (or trailing) triangular submatrix in the
              QR (or RQ) factorization with column (or row) pivoting
              whose estimated condition number is less than 1/TOL.
              If the user sets TOL to be less than SQRT((N+P)*(N+M))*EPS
              then the tolerance is taken as SQRT((N+P)*(N+M))*EPS,
              where EPS is the machine precision (see LAPACK Library
              Routine DLAMCH).

Workspace

      IWORK   INTEGER array, dimension (MAX(M,P))

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

      LDWORK  INTEGER
              The length of the array DWORK.
              LDWORK >= MAX( 1, MIN(P,M) + MAX(3*M-1,N),
                                MIN(P,N) + MAX(3*P-1,N+P,N+M),
                                MIN(M,N) + MAX(3*M-1,N+M) ).
              An upper bound is MAX(s,N) + MAX(3*s-1,N+s), with
              s = MAX(M,P).
              For optimum performance LDWORK should be larger.

              If LDWORK = -1, then a 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 related to LDWORK
              is issued by XERBLA.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value.

Method

      The routine extracts from the system matrix of a state-space
      system (A,B,C,D) a regular pencil A - lambda*B  which has the
                                         f          f
      invariant zeros of the system as generalized eigenvalues as
      follows:

         (a) construct the (N+P)-by-(N+M) compound matrix (B  A);
                                                          (D  C)

         (b) reduce the above system to one with the same invariant
             zeros and with D of full row rank;

         (c) pertranspose the system;

         (d) reduce the system to one with the same invariant zeros and
             with D square invertible;

         (e) perform a unitary transformation on the columns of
             (A - lambda*I  B) in order to reduce it to
             (      C       D)

             (A  - lambda*B   X)
             ( f           f   ), with Y and B  square invertible;
             (      0         Y)              f

         (f) compute the right and left Kronecker indices of the system
             (A,B,C,D), which together with the orders of the infinite
             zeros (determined by steps (a) - (e)) constitute the
             complete set of structural invariants under strict
             equivalence transformations of a linear system.

References

      [1] Svaricek, F.
          Computation of the Structural Invariants of Linear
          Multivariable Systems with an Extended Version of
          the Program ZEROS.
          System & Control Letters, 6, pp. 261-266, 1985.

      [2] Emami-Naeini, A. and Van Dooren, P.
          Computation of Zeros of Linear Multivariable Systems.
          Automatica, 18, pp. 415-430, 1982.

Numerical Aspects

      The algorithm is backward stable (see [2] and [1]).

Further Comments

      In order to compute the invariant zeros of the system explicitly,
      a call to this routine may be followed by a call to the LAPACK
      Library routine DGGEV with A = A , B = B  and N = NU.
                                      f       f
      If RANK = 0, the routine DGEEV can be used (since B = I).
                                                         f

Example

Program Text

    *     AB08ND EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          DOUBLE PRECISION ZERO
          PARAMETER        ( ZERO = 0.0D0 )
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX, MMAX, PMAX
          PARAMETER        ( NMAX = 20, MMAX = 20, PMAX = 20 )
          INTEGER          MPMAX
          PARAMETER        ( MPMAX = MAX( MMAX, PMAX ) )
          INTEGER          LDA, LDB, LDC, LDD, LDAF, LDBF, LDQ, LDZ
          PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
         $                   LDD = PMAX, LDAF = NMAX+MPMAX,
         $                   LDBF = NMAX+PMAX, LDQ = 1, LDZ = 1 )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = MAX( MAX( MPMAX+1, NMAX ) +
         $                                 MAX( 3*(MPMAX+1), NMAX+MPMAX ),
         $                                 8*NMAX ) )
    *     .. Local Scalars ..
          DOUBLE PRECISION TOL
          INTEGER          DINFZ, I, INFO, J, M, N, NINFZ, NKROL, NKROR,
         $                 NU, P, RANK
          CHARACTER*1      EQUIL
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), AF(LDAF,NMAX+PMAX), ALFI(NMAX),
         $                 ALFR(NMAX), B(LDB,MMAX), BETA(NMAX),
         $                 BF(LDBF,MMAX+NMAX), C(LDC,NMAX), D(LDD,MMAX),
         $                 DWORK(LDWORK), Q(LDQ,1), Z(LDZ,1)
          INTEGER          INFZ(NMAX), IWORK(MPMAX+1), KRONL(NMAX+1),
         $                 KRONR(NMAX+1)
    *     .. External Subroutines ..
          EXTERNAL         AB08ND, DGEGV
    *     .. 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, EQUIL
          IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
             WRITE ( NOUT, FMT = 99972 ) 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 = 99971 ) 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 = 99970 ) P
                ELSE
                   READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
                   READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
    *              Check the observability and compute the ordered set of
    *              the observability indices (call the routine with M = 0).
                   CALL AB08ND( EQUIL, N, 0, P, A, LDA, B, LDB, C, LDC, D,
         $                      LDD, NU, RANK, DINFZ, NKROR, NKROL, INFZ,
         $                      KRONR, KRONL, AF, LDAF, BF, LDBF, TOL,
         $                      IWORK, DWORK, LDWORK, INFO )
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      WRITE ( NOUT, FMT = 99994 ) ( KRONL(I), I = 1,P )
                      IF ( NU.EQ.0 ) THEN
                         WRITE ( NOUT, FMT = 99993 )
                      ELSE
                         WRITE ( NOUT, FMT = 99992 ) N - NU
                         WRITE ( NOUT, FMT = 99991 )
                         WRITE ( NOUT, FMT = 99990 )
                         DO 20 I = 1, NU
                            WRITE ( NOUT, FMT = 99989 )
         $                        ( AF(I,J), J = 1,NU )
       20                CONTINUE
                      END IF
                   END IF
    *              Check the controllability and compute the ordered set of
    *              the controllability indices (call the routine with P = 0)
                   CALL AB08ND( EQUIL, N, M, 0, A, LDA, B, LDB, C, LDC, D,
         $                      LDD, NU, RANK, DINFZ, NKROR, NKROL, INFZ,
         $                      KRONR, KRONL, AF, LDAF, BF, LDBF, TOL,
         $                      IWORK, DWORK, LDWORK, INFO )
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      WRITE ( NOUT, FMT = 99988 ) ( KRONR(I), I = 1,M )
                      IF ( NU.EQ.0 ) THEN
                         WRITE ( NOUT, FMT = 99987 )
                      ELSE
                         WRITE ( NOUT, FMT = 99986 ) N - NU
                         WRITE ( NOUT, FMT = 99985 )
                         WRITE ( NOUT, FMT = 99990 )
                         DO 40 I = 1, NU
                            WRITE ( NOUT, FMT = 99989 )
         $                        ( AF(I,J), J = 1,NU )
       40                CONTINUE
                      END IF
                   END IF
    *              Compute the structural invariants of the given system.
                   CALL AB08ND( EQUIL, N, M, P, A, LDA, B, LDB, C, LDC, D,
         $                      LDD, NU, RANK, DINFZ, NKROR, NKROL, INFZ,
         $                      KRONR, KRONL, AF, LDAF, BF, LDBF, TOL,
         $                      IWORK, DWORK, LDWORK, INFO )
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      WRITE ( NOUT, FMT = 99984 ) NU
                      IF ( NU.GT.0 ) THEN
    *                    Compute the invariant zeros of the given system.
    *                    Workspace: need 8*NU.
                         WRITE ( NOUT, FMT = 99983 )
                         CALL DGEGV( 'No vectors', 'No vectors', NU, AF,
         $                           LDAF, BF, LDBF, ALFR, ALFI, BETA, Q,
         $                           LDQ, Z, LDZ, DWORK, LDWORK, INFO )
    *
                         IF ( INFO.NE.0 ) THEN
                            WRITE ( NOUT, FMT = 99997 ) INFO
                         ELSE
                            WRITE ( NOUT, FMT = 99981 )
                            DO 60 I = 1, NU
                               IF ( ALFI(I).EQ.ZERO ) THEN
                                  WRITE ( NOUT, FMT = 99980 )
         $                                ALFR(I)/BETA(I)
                               ELSE
                                  WRITE ( NOUT, FMT = 99979 )
         $                                ALFR(I)/BETA(I),
         $                                ALFI(I)/BETA(I)
                               END IF
       60                   CONTINUE
                            WRITE ( NOUT, FMT = 99982 )
                         END IF
                      END IF
                      NINFZ = 0
                      DO 80 I = 1, DINFZ
                         IF ( INFZ(I).GT.0 ) THEN
                            NINFZ = NINFZ + INFZ(I)*I
                         END IF
       80             CONTINUE
                      WRITE ( NOUT, FMT = 99978 ) NINFZ
                      IF ( NINFZ.GT.0 ) THEN
                         DO 100 I = 1, DINFZ
                            WRITE ( NOUT, FMT = 99977 ) INFZ(I), I
      100                CONTINUE
                      END IF
                      WRITE ( NOUT, FMT = 99976 ) NKROR
                      IF ( NKROR.GT.0 ) WRITE ( NOUT, FMT = 99975 )
         $                                      ( KRONR(I), I = 1,NKROR )
                      WRITE ( NOUT, FMT = 99974 ) NKROL
                      IF ( NKROL.GT.0 ) WRITE ( NOUT, FMT = 99973 )
         $                                      ( KRONL(I), I = 1,NKROL )
                   END IF
                END IF
             END IF
          END IF
    *
          STOP
    *
    99999 FORMAT (' AB08ND EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from AB08ND = ',I2)
    99997 FORMAT (' INFO on exit from DGEGV = ',I2)
    99994 FORMAT (' The left Kronecker indices of (A,C) are ',/(20(I3,2X)))
    99993 FORMAT (/' The system (A,C) is completely observable ')
    99992 FORMAT (/' The dimension of the observable subspace = ',I3)
    99991 FORMAT (/' The output decoupling zeros are the eigenvalues of th',
         $       'e matrix AF. ')
    99990 FORMAT (/' The matrix AF is ')
    99989 FORMAT (20(1X,F8.4))
    99988 FORMAT (//' The right Kronecker indices of (A,B) are ',/(20(I3,2X)
         $       ))
    99987 FORMAT (/' The system (A,B) is completely controllable ')
    99986 FORMAT (/' The dimension of the controllable subspace = ',I3)
    99985 FORMAT (/' The input decoupling zeros are the eigenvalues of the',
         $       ' matrix AF. ')
    99984 FORMAT (//' The number of finite invariant zeros = ',I3)
    99983 FORMAT (/' The finite invariant zeros are ')
    99982 FORMAT (/' which correspond to the generalized eigenvalues of (l',
         $       'ambda*BF - AF).')
    99981 FORMAT (/' real  part     imag  part ')
    99980 FORMAT (1X,F9.4)
    99979 FORMAT (1X,F9.4,6X,F9.4)
    99978 FORMAT (//' The number of infinite zeros = ',I3)
    99977 FORMAT ( I4,' infinite zero(s) of order ',I3)
    99976 FORMAT (/' The number of right Kronecker indices = ',I3)
    99975 FORMAT (/' Right Kronecker (column) indices of (A,B,C,D) are ',
         $       /(20(I3,2X)))
    99974 FORMAT (/' The number of left Kronecker indices = ',I3)
    99973 FORMAT (/' The left Kronecker (row) indices of (A,B,C,D) are ',
         $       /(20(I3,2X)))
    99972 FORMAT (/' N is out of range.',/' N = ',I5)
    99971 FORMAT (/' M is out of range.',/' M = ',I5)
    99970 FORMAT (/' P is out of range.',/' P = ',I5)
          END

Program Data

     AB08ND EXAMPLE PROGRAM DATA
       6     2     3     0.0     N
       1.0   0.0   0.0   0.0   0.0   0.0
       0.0   1.0   0.0   0.0   0.0   0.0
       0.0   0.0   3.0   0.0   0.0   0.0
       0.0   0.0   0.0  -4.0   0.0   0.0
       0.0   0.0   0.0   0.0  -1.0   0.0
       0.0   0.0   0.0   0.0   0.0   3.0
       0.0  -1.0
      -1.0   0.0
       1.0  -1.0
       0.0   0.0
       0.0   1.0
      -1.0  -1.0
       1.0   0.0   0.0   1.0   0.0   0.0
       0.0   1.0   0.0   1.0   0.0   1.0
       0.0   0.0   1.0   0.0   0.0   1.0
       0.0   0.0
       0.0   0.0
       0.0   0.0

Program Results

     AB08ND EXAMPLE PROGRAM RESULTS

     The left Kronecker indices of (A,C) are
      1    2    2

     The dimension of the observable subspace =   5

     The output decoupling zeros are the eigenvalues of the matrix AF.

     The matrix AF is
      -1.0000

     The right Kronecker indices of (A,B) are
      2    3

     The dimension of the controllable subspace =   5

     The input decoupling zeros are the eigenvalues of the matrix AF.

     The matrix AF is
      -4.0000

     The number of finite invariant zeros =   2

     The finite invariant zeros are

     real  part     imag  part
        2.0000
       -1.0000

     which correspond to the generalized eigenvalues of (lambda*BF - AF).

     The number of infinite zeros =   2

     The orders of the infinite zeros are
      1    1

     The number of right Kronecker indices =   0

     The number of left Kronecker indices =   1

     The left Kronecker (row) indices of (A,B,C,D) are
      2

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