TB01VD

Conversion of a linear discrete-time system into the output normal form

Purpose

  To convert the linear discrete-time system given as (A, B, C, D),
  with initial state x0, into the output normal form [1], with
  parameter vector THETA. The matrix A is assumed to be stable.
  The matrices A, B, C, D and the vector x0 are converted, so that
  on exit they correspond to the system defined by THETA.

      SUBROUTINE TB01VD( APPLY, N, M, L, A, LDA, B, LDB, C, LDC, D, LDD,
     $                   X0, THETA, LTHETA, DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
      CHARACTER         APPLY
      INTEGER           INFO, L, LDA, LDB, LDC, LDD, LDWORK, LTHETA, M,
     $                  N
C     .. Array Arguments ..
      DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
     $                  DWORK(*), THETA(*), X0(*)

Mode Parameters

  APPLY   CHARACTER*1
          Specifies whether or not the parameter vector should be
          transformed using a bijective mapping, as follows:
          = 'A' : apply the bijective mapping to the N vectors in
                  THETA corresponding to the matrices A and C;
          = 'N' : do not apply the bijective mapping.
          The transformation performed when APPLY = 'A' allows
          to get rid of the constraints norm(THETAi) < 1, i = 1:N.
          A call of the SLICOT Library routine TB01VY associated to
          a call of TB01VD must use the same value of APPLY.

  N       (input) INTEGER
          The order of the system.  N >= 0.

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

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

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the leading N-by-N part of this array must
          contain the system state matrix A, assumed to be stable.
          On exit, the leading N-by-N part of this array contains
          the transformed system state matrix corresponding to the
          output normal form with parameter vector THETA.

  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 system input matrix B.
          On exit, the leading N-by-M part of this array contains
          the transformed system input matrix corresponding to the
          output normal form with parameter vector THETA.

  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 L-by-N part of this array must
          contain the system output matrix C.
          On exit, the leading L-by-N part of this array contains
          the transformed system output matrix corresponding to the
          output normal form with parameter vector THETA.

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

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

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

  X0      (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, this array must contain the initial state of the
          system, x0.
          On exit, this array contains the transformed initial state
          of the system, corresponding to the output normal form
          with parameter vector THETA.

  THETA   (output) DOUBLE PRECISION array, dimension (LTHETA)
          The leading N*(L+M+1)+L*M part of this array contains the
          parameter vector that defines a system (A, B, C, D, x0)
          which is equivalent up to a similarity transformation to
          the system given on entry. The parameters are:

          THETA(1:N*L)                      : parameters for A, C;
          THETA(N*L+1:N*(L+M))              : parameters for B;
          THETA(N*(L+M)+1:N*(L+M)+L*M)      : parameters for D;
          THETA(N*(L+M)+L*M+1:N*(L+M+1)+L*M): parameters for x0.

  LTHETA  INTEGER
          The length of array THETA.  LTHETA >= N*(L+M+1)+L*M.

  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, N*N*L + N*L + N,
                        N*N + MAX(N*N + N*MAX(N,L) + 6*N + MIN(N,L),
                                  N*M)).
          For optimum performance LDWORK should be larger.

  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value;
          = 1:  if the Lyapunov equation A'*Q*A - Q = -scale^2*C'*C
                could only be solved with scale = 0;
          = 2:  if matrix A is not discrete-time stable;
          = 3:  if the QR algorithm failed to converge for
                matrix A.

  The matrices A and C are converted to output normal form.
  First, the Lyapunov equation

     A'*Q*A - Q = -scale^2*C'*C,

  is solved in the Cholesky factor T, T'*T = Q, and then T is used
  to get the transformation matrix.

  The matrix B and the initial state x0 are transformed accordingly.

  Then, the QR factorization of the transposed observability matrix
  is computed, and the matrix Q is used to further transform the
  system matrices. The parameters characterizing A and C are finally
  obtained by applying a set of N orthogonal transformations.

  [1] Peeters, R.L.M., Hanzon, B., and Olivi, M.
      Balanced realizations of discrete-time stable all-pass
      systems and the tangential Schur algorithm.
      Proceedings of the European Control Conference,
      31 August - 3 September 1999, Karlsruhe, Germany.
      Session CP-6, Discrete-time Systems, 1999.

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