TC04AD

State-space representation for a given left/right polynomial matrix representation

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

Purpose

  To find a state-space representation (A,B,C,D) with the same
  transfer matrix T(s) as that of a given left or right polynomial
  matrix representation, i.e.

     C*inv(sI-A)*B + D = T(s) = inv(P(s))*Q(s) = Q(s)*inv(P(s)).

Specification
      SUBROUTINE TC04AD( LERI, M, P, INDEX, PCOEFF, LDPCO1, LDPCO2,
     $                   QCOEFF, LDQCO1, LDQCO2, N, RCOND, A, LDA, B,
     $                   LDB, C, LDC, D, LDD, IWORK, DWORK, LDWORK,
     $                   INFO )
C     .. Scalar Arguments ..
      CHARACTER         LERI
      INTEGER           INFO, LDA, LDB, LDC, LDD, LDPCO1, LDPCO2,
     $                  LDQCO1, LDQCO2, LDWORK, M, N, P
      DOUBLE PRECISION  RCOND
C     .. Array Arguments ..
      INTEGER           INDEX(*), IWORK(*)
      DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
     $                  DWORK(*), PCOEFF(LDPCO1,LDPCO2,*),
     $                  QCOEFF(LDQCO1,LDQCO2,*)

Arguments

Mode Parameters

  LERI    CHARACTER*1
          Indicates whether a left polynomial matrix representation
          or a right polynomial matrix representation is input as
          follows:
          = 'L':  A left matrix fraction is input;
          = 'R':  A right matrix fraction is input.

Input/Output Parameters
  M       (input) INTEGER
          The number of system inputs.  M >= 0.

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

  INDEX   (input) INTEGER array, dimension (MAX(M,P))
          If LERI = 'L', INDEX(I), I = 1,2,...,P, must contain the
          maximum degree of the polynomials in the I-th row of the
          denominator matrix P(s) of the given left polynomial
          matrix representation.
          If LERI = 'R', INDEX(I), I = 1,2,...,M, must contain the
          maximum degree of the polynomials in the I-th column of
          the denominator matrix P(s) of the given right polynomial
          matrix representation.

  PCOEFF  (input) DOUBLE PRECISION array, dimension
          (LDPCO1,LDPCO2,kpcoef), where kpcoef = MAX(INDEX(I)) + 1.
          If LERI = 'L' then porm = P, otherwise porm = M.
          The leading porm-by-porm-by-kpcoef part of this array must
          contain the coefficients of the denominator matrix P(s).
          PCOEFF(I,J,K) is the coefficient in s**(INDEX(iorj)-K+1)
          of polynomial (I,J) of P(s), where K = 1,2,...,kpcoef; if
          LERI = 'L' then iorj = I, otherwise iorj = J.
          Thus for LERI = 'L', P(s) =
          diag(s**INDEX(I))*(PCOEFF(.,.,1)+PCOEFF(.,.,2)/s+...).
          If LERI = 'R', PCOEFF is modified by the routine but
          restored on exit.

  LDPCO1  INTEGER
          The leading dimension of array PCOEFF.
          LDPCO1 >= MAX(1,P) if LERI = 'L',
          LDPCO1 >= MAX(1,M) if LERI = 'R'.

  LDPCO2  INTEGER
          The second dimension of array PCOEFF.
          LDPCO2 >= MAX(1,P) if LERI = 'L',
          LDPCO2 >= MAX(1,M) if LERI = 'R'.

  QCOEFF  (input) DOUBLE PRECISION array, dimension
          (LDQCO1,LDQCO2,kpcoef)
          If LERI = 'L' then porp = M, otherwise porp = P.
          The leading porm-by-porp-by-kpcoef part of this array must
          contain the coefficients of the numerator matrix Q(s).
          QCOEFF(I,J,K) is defined as for PCOEFF(I,J,K).
          If LERI = 'R', QCOEFF is modified by the routine but
          restored on exit.

  LDQCO1  INTEGER
          The leading dimension of array QCOEFF.
          LDQCO1 >= MAX(1,P)   if LERI = 'L',
          LDQCO1 >= MAX(1,M,P) if LERI = 'R'.

  LDQCO2  INTEGER
          The second dimension of array QCOEFF.
          LDQCO2 >= MAX(1,M)   if LERI = 'L',
          LDQCO2 >= MAX(1,M,P) if LERI = 'R'.

  N       (output) INTEGER
          The order of the resulting state-space representation.
                       porm
          That is, N = SUM INDEX(I).
                       I=1

  RCOND   (output) DOUBLE PRECISION
          The estimated reciprocal of the condition number of the
          leading row (if LERI = 'L') or the leading column (if
          LERI = 'R') coefficient matrix of P(s).
          If RCOND is nearly zero, P(s) is nearly row or column
          non-proper.

  A       (output) DOUBLE PRECISION array, dimension (LDA,N)
          The leading N-by-N part of this array contains the state
          dynamics matrix A.

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

  B       (output) DOUBLE PRECISION array, dimension (LDB,MAX(M,P))
          The leading N-by-M part of this array contains the
          input/state matrix B; the remainder of the leading
          N-by-MAX(M,P) part is used as internal workspace.

  LDB     INTEGER
          The leading dimension of array B.  LDB >= MAX(1,N).

  C       (output) DOUBLE PRECISION array, dimension (LDC,N)
          The leading P-by-N part of this array contains the
          state/output matrix C; the remainder of the leading
          MAX(M,P)-by-N part is used as internal workspace.

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

  D       (output) DOUBLE PRECISION array, dimension (LDD,MAX(M,P))
          The leading P-by-M part of this array contains the direct
          transmission matrix D; the remainder of the leading
          MAX(M,P)-by-MAX(M,P) part is used as internal workspace.

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

Workspace
  IWORK   INTEGER array, dimension (2*MAX(M,P))

  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,MAX(M,P)*(MAX(M,P)+4)).
          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 P(s) is not row (if LERI = 'L') or column
                (if LERI = 'R') proper. Consequently, no state-space
                representation is calculated.

Method
  The method for a left matrix fraction will be described here;
  right matrix fractions are dealt with by obtaining the dual left
  polynomial matrix representation and constructing an equivalent
  state-space representation for this. The first step is to check
  if the denominator matrix P(s) is row proper; if it is not then
  the routine returns with the Error Indicator (INFO) set to 1.
  Otherwise, Wolovich's Observable  Structure Theorem is used to
  construct a state-space representation (A,B,C,D) in observable
  companion form. The sizes of the blocks of matrix A and matrix C
  here are precisely the row degrees of P(s), while their
  'non-trivial' columns are given easily from its coefficients.
  Similarly, the matrix D is obtained from the leading coefficients
  of P(s) and of the numerator matrix Q(s), while matrix B is given
  by the relation Sbar(s)B = Q(s) - P(s)D, where Sbar(s) is a
  polynomial matrix whose (j,k)(th) element is given by

               j-u(k-1)-1
            ( s           , j = u(k-1)+1,u(k-1)+2,....,u(k)
  Sbar    = (
     j,k    (           0 , otherwise

          k
  u(k) = SUM d , k = 1,2,...,M and d ,d ,...,d  are the
         i=1  i                     1  2      M
  controllability indices. For convenience in solving this, C' and B
  are initially set up to contain the coefficients of P(s) and Q(s),
  respectively, stored by rows.

References
  [1] Wolovich, W.A.
      Linear Multivariate Systems, (Theorem 4.3.3).
      Springer-Verlag, 1974.

Numerical Aspects
                            3
  The algorithm requires 0(N ) operations.

Further Comments
  None
Example

Program Text

*     TC04AD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
      INTEGER          NIN, NOUT
      PARAMETER        ( NIN = 5, NOUT = 6 )
      INTEGER          MMAX, PMAX, KPCMAX, NMAX
      PARAMETER        ( MMAX = 5, PMAX = 5, KPCMAX = 5, NMAX = 5 )
      INTEGER          MAXMP
      PARAMETER        ( MAXMP = MAX( MMAX, PMAX ) )
      INTEGER          LDPCO1, LDPCO2, LDQCO1, LDQCO2, LDA, LDB, LDC,
     $                 LDD
      PARAMETER        ( LDPCO1 = MAXMP, LDPCO2 = MAXMP,
     $                   LDQCO1 = MAXMP, LDQCO2 = MAXMP,
     $                   LDA = NMAX, LDB = NMAX, LDC = MAXMP,
     $                   LDD = MAXMP )
      INTEGER          LIWORK
      PARAMETER        ( LIWORK = 2*MAXMP )
      INTEGER          LDWORK
      PARAMETER        ( LDWORK = ( MAXMP )*( MAXMP+4 ) )
*     .. Local Scalars ..
      DOUBLE PRECISION RCOND
      INTEGER          I, INFO, J, K, KPCOEF, M, N, P, PORM, PORP
      CHARACTER*1      LERI
      LOGICAL          LLERI
*     .. Local Arrays ..
      DOUBLE PRECISION A(LDA,NMAX), B(LDB,MAXMP), C(LDC,NMAX),
     $                 D(LDD,MAXMP), PCOEFF(LDPCO1,LDPCO2,KPCMAX),
     $                 QCOEFF(LDQCO1,LDQCO2,KPCMAX), DWORK(LDWORK)
      INTEGER          INDEX(MAXMP), IWORK(LIWORK)
*     .. External Functions ..
      LOGICAL          LSAME
      EXTERNAL         LSAME
*     .. External Subroutines ..
      EXTERNAL         TC04AD
*     .. 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 = * ) M, P, LERI
      LLERI = LSAME( LERI, 'L' )
      IF ( M.LE.0 .OR. M.GT.MMAX ) THEN
         WRITE ( NOUT, FMT = 99991 ) M
      ELSE IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
         WRITE ( NOUT, FMT = 99990 ) P
      ELSE
         PORM = P
         IF ( .NOT.LLERI ) PORM = M
         READ ( NIN, FMT = * ) ( INDEX(I), I = 1,PORM )
         PORP = M
         IF ( .NOT.LLERI ) PORP = P
         KPCOEF = 0
         DO 20 I = 1, PORM
            KPCOEF = MAX( KPCOEF, INDEX(I) )
   20    CONTINUE
         KPCOEF = KPCOEF + 1
         IF ( KPCOEF.LE.0 .OR. KPCOEF.GT.KPCMAX ) THEN
            WRITE ( NOUT, FMT = 99989 ) KPCOEF
         ELSE
            READ ( NIN, FMT = * )
     $         ( ( ( PCOEFF(I,J,K), K = 1,KPCOEF ), J = 1,PORM ),
     $                              I = 1,PORM )
            READ ( NIN, FMT = * )
     $         ( ( ( QCOEFF(I,J,K), K = 1,KPCOEF ), J = 1,PORP ),
     $                              I = 1,PORM )
*           Find a ssr of the given left pmr.
            CALL TC04AD( LERI, M, P, INDEX, PCOEFF, LDPCO1, LDPCO2,
     $                   QCOEFF, LDQCO1, LDQCO2, N, RCOND, A, LDA, B,
     $                   LDB, C, LDC, D, LDD, IWORK, DWORK, LDWORK,
     $                   INFO )
*
            IF ( INFO.NE.0 ) THEN
               WRITE ( NOUT, FMT = 99998 ) INFO
            ELSE
               WRITE ( NOUT, FMT = 99997 ) N, RCOND
               WRITE ( NOUT, FMT = 99996 )
               DO 40 I = 1, N
                  WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,N )
   40          CONTINUE
               WRITE ( NOUT, FMT = 99994 )
               DO 60 I = 1, N
                  WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
   60          CONTINUE
               WRITE ( NOUT, FMT = 99993 )
               DO 80 I = 1, P
                  WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,N )
   80          CONTINUE
               WRITE ( NOUT, FMT = 99992 )
               DO 100 I = 1, P
                  WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1,M )
  100          CONTINUE
            END IF
         END IF
      END IF
      STOP
*
99999 FORMAT (' TC04AD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TC04AD = ',I2)
99997 FORMAT (' The order of the resulting state-space representation ',
     $       ' =  ',I2,//' RCOND = ',F4.2)
99996 FORMAT (/' The state dynamics matrix A is ')
99995 FORMAT (20(1X,F8.4))
99994 FORMAT (/' The input/state matrix B is ')
99993 FORMAT (/' The state/output matrix C is ')
99992 FORMAT (/' The direct transmission matrix D is ')
99991 FORMAT (/' M is out of range.',/' M = ',I5)
99990 FORMAT (/' P is out of range.',/' P = ',I5)
99989 FORMAT (/' KPCOEF is out of range.',/' KPCOEF = ',I5)
      END
Program Data
 TC04AD EXAMPLE PROGRAM DATA
   2     2     L
   2     2
   2.0   3.0   1.0
   4.0  -1.0  -1.0
   5.0   7.0  -6.0
   3.0   2.0   2.0
   6.0  -1.0   5.0
   1.0   7.0   5.0
   1.0   1.0   1.0
   4.0   1.0  -1.0
Program Results
 TC04AD EXAMPLE PROGRAM RESULTS

 The order of the resulting state-space representation  =   4

 RCOND = 0.25

 The state dynamics matrix A is 
   0.0000   0.5714   0.0000  -0.4286
   1.0000   1.0000   0.0000  -1.0000
   0.0000  -2.0000   0.0000   2.0000
   0.0000   0.7857   1.0000  -1.7143

 The input/state matrix B is 
   8.0000   3.8571
   4.0000   4.0000
  -9.0000   5.0000
   4.0000  -5.0714

 The state/output matrix C is 
   0.0000  -0.2143   0.0000   0.2857
   0.0000   0.3571   0.0000  -0.1429

 The direct transmission matrix D is 
  -1.0000   0.9286
   2.0000  -0.2143

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