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
NoneExample
Program Text
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