SB08FD

Right coprime factorization with prescribed stability degree

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

Purpose

  To construct, for a given system G = (A,B,C,D), a feedback
  matrix F and an orthogonal transformation matrix Z, such that
  the systems

       Q = (Z'*(A+B*F)*Z, Z'*B, (C+D*F)*Z, D)
  and
       R = (Z'*(A+B*F)*Z, Z'*B, F*Z, I)

  provide a stable right coprime factorization of G in the form
                    -1
           G = Q * R  ,

  where G, Q and R are the corresponding transfer-function matrices.
  The resulting state dynamics matrix of the systems Q and R has
  eigenvalues lying inside a given stability domain.
  The Z matrix is not explicitly computed.

  Note: If the given state-space representation is not stabilizable,
  the unstabilizable part of the original system is automatically
  deflated and the order of the systems Q and R is accordingly
  reduced.

Specification
      SUBROUTINE SB08FD( DICO, N, M, P, ALPHA, A, LDA, B, LDB, C, LDC,
     $                   D, LDD, NQ, NR, CR, LDCR, DR, LDDR, TOL, DWORK,
     $                   LDWORK, IWARN, INFO )
C     .. Scalar Arguments ..
      CHARACTER         DICO
      INTEGER           INFO, IWARN, LDA, LDB, LDC, LDCR, LDD, LDDR,
     $                  LDWORK, M, N, NQ, NR, P
      DOUBLE PRECISION  TOL
C     .. Array Arguments ..
      DOUBLE PRECISION  A(LDA,*), ALPHA(*), B(LDB,*), C(LDC,*),
     $                  CR(LDCR,*), D(LDD,*), DR(LDDR,*), DWORK(*)

Arguments

Mode Parameters

  DICO    CHARACTER*1
          Specifies the type of the original system as follows:
          = 'C':  continuous-time system;
          = 'D':  discrete-time system.

Input/Output Parameters
  N       (input) INTEGER
          The dimension of the state vector, i.e. the order of the
          matrix A, and also the number of rows of the matrix B and
          the number of columns of the matrices C and CR.  N >= 0.

  M       (input) INTEGER
          The dimension of input vector, i.e. the number of columns
          of the matrices B, D and DR and the number of rows of the
          matrices CR and DR.  M >= 0.

  P       (input) INTEGER
          The dimension of output vector, i.e. the number of rows
          of the matrices C and D.  P >= 0.

  ALPHA   (input) DOUBLE PRECISION array, dimension (2)
          ALPHA(1) contains the desired stability degree to be
          assigned for the eigenvalues of A+B*F, and ALPHA(2)
          the stability margin. The eigenvalues outside the
          ALPHA(2)-stability region will be assigned to have the
          real parts equal to ALPHA(1) < 0 and unmodified
          imaginary parts for a continuous-time system
          (DICO = 'C'), or moduli equal to 0 <= ALPHA(2) < 1
          for a discrete-time system (DICO = 'D').

  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
          On entry, the leading N-by-N part of this array must
          contain the state dynamics matrix A.
          On exit, the leading NQ-by-NQ part of this array contains
          the leading NQ-by-NQ part of the matrix Z'*(A+B*F)*Z, the
          state dynamics matrix of the numerator factor Q, in a
          real Schur form. The trailing NR-by-NR part of this matrix
          represents the state dynamics matrix of a minimal
          realization of the denominator factor R.

  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 input/state matrix.
          On exit, the leading NQ-by-M part of this array contains
          the leading NQ-by-M part of the matrix Z'*B, the
          input/state matrix of the numerator factor Q. The last
          NR rows of this matrix form the input/state matrix of
          a minimal realization of the denominator factor R.

  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 P-by-N part of this array must
          contain the state/output matrix C.
          On exit, the leading P-by-NQ part of this array contains
          the leading P-by-NQ part of the matrix (C+D*F)*Z,
          the state/output matrix of the numerator factor Q.

  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
          input/output matrix. D represents also the input/output
          matrix of the numerator factor Q.

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

  NQ      (output) INTEGER
          The order of the resulting factors Q and R.
          Generally, NQ = N - NS, where NS is the number of
          uncontrollable eigenvalues outside the stability region.

  NR      (output) INTEGER
          The order of the minimal realization of the factor R.
          Generally, NR is the number of controllable eigenvalues
          of A outside the stability region (the number of modified
          eigenvalues).

  CR      (output) DOUBLE PRECISION array, dimension (LDCR,N)
          The leading M-by-NQ part of this array contains the
          leading M-by-NQ part of the feedback matrix F*Z, which
          moves the eigenvalues of A lying outside the ALPHA-stable
          region to values which are on the ALPHA-stability
          boundary.  The last NR columns of this matrix form the
          state/output matrix of a minimal realization of the
          denominator factor R.

  LDCR    INTEGER
          The leading dimension of array CR.  LDCR >= MAX(1,M).

  DR      (output) DOUBLE PRECISION array, dimension (LDDR,M)
          The leading M-by-M part of this array contains an
          identity matrix representing the input/output matrix
          of the denominator factor R.

  LDDR    INTEGER
          The leading dimension of array DR.  LDDR >= MAX(1,M).

Tolerances
  TOL     DOUBLE PRECISION
          The absolute tolerance level below which the elements of
          B are considered zero (used for controllability tests).
          If the user sets TOL <= 0, then an implicitly computed,
          default tolerance, defined by  TOLDEF = N*EPS*NORM(B),
          is used instead, where EPS is the machine precision
          (see LAPACK Library routine DLAMCH) and NORM(B) denotes
          the 1-norm of B.

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

  LDWORK  INTEGER
          The dimension of working array DWORK.
          LWORK >= MAX( 1, N*(N+5), 5*M, 4*P ).
          For optimum performance LDWORK should be larger.

Warning Indicator
  IWARN   INTEGER
          = 0:  no warning;
          = K:  K violations of the numerical stability condition
                NORM(F) <= 10*NORM(A)/NORM(B) occured during the
                assignment of eigenvalues.

Error Indicator
  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value;
          = 1:  the reduction of A to a real Schur form failed;
          = 2:  a failure was detected during the ordering of the
                real Schur form of A, or in the iterative process
                for reordering the eigenvalues of Z'*(A + B*F)*Z
                along the diagonal.

Method
  The subroutine is based on the factorization algorithm of [1].

References
  [1] Varga A.
      Coprime factors model reduction method based on
      square-root balancing-free techniques.
      System Analysis, Modelling and Simulation,
      vol. 11, pp. 303-311, 1993.

Numerical Aspects
                                         3
  The algorithm requires no more than 14N  floating point
  operations.

Further Comments
  None
Example

Program Text

*     SB08FD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
      INTEGER          NIN, NOUT
      PARAMETER        ( NIN = 5, NOUT = 6 )
      INTEGER          NMAX, MMAX, PMAX
      PARAMETER        ( NMAX = 20, MMAX = 20, PMAX = 20 )
      INTEGER          LDA, LDB, LDC, LDCR, LDD, LDDR
      PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
     $                   LDCR = MMAX, LDD = PMAX, LDDR = MMAX )
      INTEGER          LDWORK
      PARAMETER        ( LDWORK = MAX( NMAX*( NMAX + 5 ), 5*MMAX,
     $                                 4*PMAX ) )
*     .. Local Scalars ..
      DOUBLE PRECISION TOL
      INTEGER          I, INFO, IWARN, J, M, N, NQ, NR, P
      CHARACTER*1      DICO
*     .. Local Arrays ..
      DOUBLE PRECISION A(LDA,NMAX), ALPHA(2), B(LDB,MMAX), C(LDC,NMAX),
     $                 CR(LDCR,NMAX), D(LDD,MMAX), DR(LDDR,MMAX),
     $                 DWORK(LDWORK)
*     .. External Subroutines ..
      EXTERNAL         SB08FD
*     .. 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, ALPHA(1), TOL, DICO
      ALPHA(2) = ALPHA(1)
      IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
         WRITE ( NOUT, FMT = 99990 ) 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 = 99989 ) 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 = 99988 ) 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 )
*              Find a RCF for (A,B,C,D).
               CALL SB08FD( DICO, N, M, P, ALPHA, A, LDA, B, LDB, C,
     $                      LDC, D, LDD, NQ, NR, CR, LDCR, DR, LDDR,
     $                      TOL, DWORK, LDWORK, IWARN, INFO )
*
               IF ( INFO.NE.0 ) THEN
                  WRITE ( NOUT, FMT = 99998 ) INFO
               ELSE
                  IF( NQ.GT.0 ) WRITE ( NOUT, FMT = 99996 )
                  DO 20 I = 1, NQ
                     WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1, NQ )
   20             CONTINUE
                  IF( NQ.GT.0 ) WRITE ( NOUT, FMT = 99993 )
                  DO 40 I = 1, NQ
                     WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1, M )
   40             CONTINUE
                  IF( NQ.GT.0 ) WRITE ( NOUT, FMT = 99992 )
                  DO 60 I = 1, P
                     WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1, NQ )
   60             CONTINUE
                  WRITE ( NOUT, FMT = 99991 )
                  DO 70 I = 1, P
                     WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1, M )
   70             CONTINUE
                  IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99986 )
                  DO 80 I = NQ-NR+1, NQ
                     WRITE ( NOUT, FMT = 99995 )
     $                     ( A(I,J), J = NQ-NR+1, NQ )
   80             CONTINUE
                  IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99985 )
                  DO 90 I = NQ-NR+1, NQ
                     WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1, M )
   90             CONTINUE
                  IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99984 )
                  DO 100 I = 1, M
                     WRITE ( NOUT, FMT = 99995 )
     $                     ( CR(I,J), J = NQ-NR+1, NQ )
  100             CONTINUE
                  WRITE ( NOUT, FMT = 99983 )
                  DO 110 I = 1, M
                     WRITE ( NOUT, FMT = 99995 ) ( DR(I,J), J = 1, M )
  110             CONTINUE
               END IF
            END IF
         END IF
      END IF
      STOP
*
99999 FORMAT (' SB08FD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB08FD = ',I2)
99996 FORMAT (/' The numerator state dynamics matrix AQ is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The numerator input/state matrix BQ is ')
99992 FORMAT (/' The numerator state/output matrix CQ is ')
99991 FORMAT (/' The numerator input/output matrix DQ is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99986 FORMAT (/' The denominator state dynamics matrix AR is ')
99985 FORMAT (/' The denominator input/state matrix BR is ')
99984 FORMAT (/' The denominator state/output matrix CR is ')
99983 FORMAT (/' The denominator input/output matrix DR is ')
      END
Program Data
 SB08FD EXAMPLE PROGRAM DATA (Continuous system)
  7  2  3 -1.0   1.E-10 C
 -0.04165  0.0000  4.9200   0.4920  0.0000   0.0000  0.0000
 -5.2100  -12.500  0.0000   0.0000  0.0000   0.0000  0.0000
  0.0000   3.3300 -3.3300   0.0000  0.0000   0.0000  0.0000
  0.5450   0.0000  0.0000   0.0000  0.0545   0.0000  0.0000
  0.0000   0.0000  0.0000  -0.49200 0.004165 0.0000  4.9200
  0.0000   0.0000  0.0000   0.0000  0.5210  -12.500  0.0000
  0.0000   0.0000  0.0000   0.0000  0.0000   3.3300 -3.3300
  0.0000   0.0000
  12.500   0.0000
  0.0000   0.0000
  0.0000   0.0000
  0.0000   0.0000
  0.0000   12.500
  0.0000   0.0000
  1.0000   0.0000  0.0000   0.0000  0.0000  0.0000  0.0000
  0.0000   0.0000  0.0000   1.0000  0.0000  0.0000  0.0000
  0.0000   0.0000  0.0000   0.0000  1.0000  0.0000  0.0000
  0.0000   0.0000  
  0.0000   0.0000  
  0.0000   0.0000  
Program Results
 SB08FD EXAMPLE PROGRAM RESULTS


 The numerator state dynamics matrix AQ is 
  -1.4178  -5.1682   3.2450  -0.2173   0.0564  -3.2129  -3.6183
   0.9109  -1.4178  -2.1262   0.1231   0.0805  -0.4392  -0.2528
   0.0000   0.0000 -13.1627   0.0608  -0.0218   2.3461   5.8272
   0.0000   0.0000   0.0000  -3.5957  -3.3373   1.3622  -3.6083
   0.0000   0.0000   0.0000   0.0000 -12.4245  -9.8634   8.1191
   0.0000   0.0000   0.0000   0.0000   0.0000  -1.0000  -0.0135
   0.0000   0.0000   0.0000   0.0000   0.0000   1.7393  -1.0000

 The numerator input/state matrix BQ is 
   5.0302  -0.0063
   0.7078  -0.0409
 -11.3663   0.0051
   0.1760   0.5879
  -0.0265  12.2119
   1.0104   1.3262
   0.4474  -2.2388

 The numerator state/output matrix CQ is 
  -0.8659   0.2787  -0.3432   0.0020   0.0000   0.2026   0.1172
   0.0797  -0.3951   0.0976  -0.0292   0.0062   0.7676   0.4879
  -0.0165  -0.0645   0.0097   0.8032  -0.1602   0.3050  -0.4812

 The numerator input/output matrix DQ is 
   0.0000   0.0000
   0.0000   0.0000
   0.0000   0.0000

 The denominator state dynamics matrix AR is 
  -1.0000  -0.0135
   1.7393  -1.0000

 The denominator input/state matrix BR is 
   1.0104   1.3262
   0.4474  -2.2388

 The denominator state/output matrix CR is 
  -0.1091  -0.4653
  -0.7055   0.4766

 The denominator input/output matrix DR is 
   1.0000   0.0000
   0.0000   1.0000

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