SB10ZP

Transforming a SISO system into a stable and minimum phase one

Purpose

  To transform a SISO (single-input single-output) system [A,B;C,D]
  by mirroring its unstable poles and zeros in the boundary of the
  stability domain, thus preserving the frequency response of the
  system, but making it stable and minimum phase. Specifically, for
  a continuous-time system, the positive real parts of its poles
  and zeros are exchanged with their negatives. Discrete-time
  systems are first converted to continuous-time systems using a
  bilinear transformation, and finally converted back.

      SUBROUTINE SB10ZP( DISCFL, N, A, LDA, B, C, D, IWORK, DWORK,
     $                   LDWORK, INFO )
C     .. Scalar Arguments ..
      INTEGER            DISCFL, INFO, LDA, LDWORK, N
C     .. Array Arguments ..
      INTEGER            IWORK( * )
      DOUBLE PRECISION   A( LDA, * ), B( * ), C( * ), D( * ), DWORK( * )

Input/Output Parameters

  DISCFL  (input) INTEGER
          Indicates the type of the system, as follows:
          = 0: continuous-time system;
          = 1: discrete-time system.

  N       (input/output) INTEGER
          On entry, the order of the original system.  N >= 0.
          On exit, the order of the transformed, minimal system.

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the leading N-by-N part of this array must
          contain the original system matrix A.
          On exit, the leading N-by-N part of this array contains
          the transformed matrix A, in an upper Hessenberg form.

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

  B       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, this array must contain the original system
          vector B.
          On exit, this array contains the transformed vector B.

  C       (input/output) DOUBLE PRECISION array, dimension (N)
          On entry, this array must contain the original system
          vector C.
          On exit, this array contains the transformed vector C.
          The first N-1 elements are zero (for the exit value of N).

  D       (input/output) DOUBLE PRECISION array, dimension (1)
          On entry, this array must contain the original system
          scalar D.
          On exit, this array contains the transformed scalar D.

  IWORK   INTEGER array, dimension (max(2,N+1))

  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(N*N + 5*N, 6*N + 1 + min(1,N)).
          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 discrete --> continuous transformation cannot
                be made;
          = 2:  if the system poles cannot be found;
          = 3:  if the inverse system cannot be found, i.e., D is
                (close to) zero;
          = 4:  if the system zeros cannot be found;
          = 5:  if the state-space representation of the new
                transfer function T(s) cannot be found;
          = 6:  if the continuous --> discrete transformation cannot
                be made.

  First, if the system is discrete-time, it is transformed to
  continuous-time using alpha = beta = 1 in the bilinear
  transformation implemented in the SLICOT routine AB04MD.
  Then the eigenvalues of A, i.e., the system poles, are found.
  Then, the inverse of the original system is found and its poles,
  i.e., the system zeros, are evaluated.
  The obtained system poles Pi and zeros Zi are checked and if a
  positive real part is detected, it is exchanged by -Pi or -Zi.
  Then the polynomial coefficients of the transfer function
  T(s) = Q(s)/P(s) are found.
  The state-space representation of T(s) is then obtained.
  The system matrices B, C, D are scaled so that the transformed
  system has the same system gain as the original system.
  If the original system is discrete-time, then the result (which is
  continuous-time) is converted back to discrete-time.

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Program Text

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