## SB10ZP

### Transforming a SISO system into a stable and minimum phase one

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

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.

Specification

          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( * )

Arguments

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.

Workspace

      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.

Error Indicator

      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.

Method

      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.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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