## SB16BD

### Coprime factorization based state feedback controller reduction

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

Purpose

      To compute, for a given open-loop model (A,B,C,D), and for
      given state feedback gain F and full observer gain G,
      such that A+B*F and A+G*C are stable, a reduced order
      controller model (Ac,Bc,Cc,Dc) using a coprime factorization
      based controller reduction approach. For reduction,
      either the square-root or the balancing-free square-root
      versions of the Balance & Truncate (B&T) or Singular Perturbation
      Approximation (SPA) model reduction methods are used in
      conjunction with stable coprime factorization techniques.

Specification

          SUBROUTINE SB16BD( DICO, JOBD, JOBMR, JOBCF, EQUIL, ORDSEL,
         $                   N, M, P, NCR, A, LDA, B, LDB, C, LDC, D, LDD,
         $                   F, LDF, G, LDG, DC, LDDC, HSV, TOL1, TOL2,
         $                   IWORK, DWORK, LDWORK, IWARN, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         DICO, EQUIL, JOBCF, JOBD, JOBMR, ORDSEL
          INTEGER           INFO, IWARN, LDA, LDB, LDC, LDD, LDDC,
         $                  LDF, LDG, LDWORK, M, N, NCR, P
          DOUBLE PRECISION  TOL1, TOL2
    C     .. Array Arguments ..
          INTEGER           IWORK(*)
          DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
         $                  DC(LDDC,*), DWORK(*), F(LDF,*), G(LDG,*), HSV(*)

Arguments

Mode Parameters

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

      JOBD    CHARACTER*1
              Specifies whether or not a non-zero matrix D appears
              in the given state space model:
              = 'D':  D is present;
              = 'Z':  D is assumed a zero matrix.

      JOBMR   CHARACTER*1
              Specifies the model reduction approach to be used
              as follows:
              = 'B':  use the square-root B&T method;
              = 'F':  use the balancing-free square-root B&T method;
              = 'S':  use the square-root SPA method;
              = 'P':  use the balancing-free square-root SPA method.

      JOBCF   CHARACTER*1
              Specifies whether left or right coprime factorization is
              to be used as follows:
              = 'L':  use left coprime factorization;
              = 'R':  use right coprime factorization.

      EQUIL   CHARACTER*1
              Specifies whether the user wishes to perform a
              preliminary equilibration before performing
              order reduction as follows:
              = 'S':  perform equilibration (scaling);
              = 'N':  do not perform equilibration.

      ORDSEL  CHARACTER*1
              Specifies the order selection method as follows:
              = 'F':  the resulting controller order NCR is fixed;
              = 'A':  the resulting controller order NCR is
                      automatically determined on basis of the given
                      tolerance TOL1.

Input/Output Parameters

      N       (input) INTEGER
              The order of the open-loop state-space representation,
              i.e., the order of the matrix A.  N >= 0.
              N also represents the order of the original state-feedback
              controller.

      M       (input) INTEGER
              The number of system inputs.  M >= 0.

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

      NCR     (input/output) INTEGER
              On entry with ORDSEL = 'F', NCR is the desired order of
              the resulting reduced order controller.  0 <= NCR <= N.
              On exit, if INFO = 0, NCR is the order of the resulting
              reduced order controller. NCR is set as follows:
              if ORDSEL = 'F', NCR is equal to MIN(NCR,NMIN), where NCR
              is the desired order on entry, and NMIN is the order of a
              minimal realization of an extended system Ge (see METHOD);
              NMIN is determined as the number of
              Hankel singular values greater than N*EPS*HNORM(Ge),
              where EPS is the machine precision (see LAPACK Library
              Routine DLAMCH) and HNORM(Ge) is the Hankel norm of the
              extended system (computed in HSV(1));
              if ORDSEL = 'A', NCR is equal to the number of Hankel
              singular values greater than MAX(TOL1,N*EPS*HNORM(Ge)).

      A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
              On entry, the leading N-by-N part of this array must
              contain the original state dynamics matrix A.
              On exit, if INFO = 0, the leading NCR-by-NCR part of this
              array contains the state dynamics matrix Ac of the reduced
              controller.

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

      B       (input) DOUBLE PRECISION array, dimension (LDB,M)
              The leading N-by-M part of this array must
              contain the original input/state matrix B.

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

      C       (input) DOUBLE PRECISION array, dimension (LDC,N)
              The leading P-by-N part of this array must
              contain the original state/output matrix C.

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

      D       (input) DOUBLE PRECISION array, dimension (LDD,M)
              If JOBD = 'D', the leading P-by-M part of this
              array must contain the system direct input/output
              transmission matrix D.
              The array D is not referenced if JOBD = 'Z'.

      LDD     INTEGER
              The leading dimension of array D.
              LDD >= MAX(1,P), if JOBD = 'D';
              LDD >= 1,        if JOBD = 'Z'.

      F       (input/output) DOUBLE PRECISION array, dimension (LDF,N)
              On entry, the leading M-by-N part of this array must
              contain a stabilizing state feedback matrix.
              On exit, if INFO = 0, the leading M-by-NCR part of this
              array contains the state/output matrix Cc of the reduced
              controller.

      LDF     INTEGER
              The leading dimension of array F.  LDF >= MAX(1,M).

      G       (input/output) DOUBLE PRECISION array, dimension (LDG,P)
              On entry, the leading N-by-P part of this array must
              contain a stabilizing observer gain matrix.
              On exit, if INFO = 0, the leading NCR-by-P part of this
              array contains the input/state matrix Bc of the reduced
              controller.

      LDG     INTEGER
              The leading dimension of array G.  LDG >= MAX(1,N).

      DC      (output) DOUBLE PRECISION array, dimension (LDDC,P)
              If INFO = 0, the leading M-by-P part of this array
              contains the input/output matrix Dc of the reduced
              controller.

      LDDC    INTEGER
              The leading dimension of array DC.  LDDC >= MAX(1,M).

      HSV     (output) DOUBLE PRECISION array, dimension (N)
              If INFO = 0, it contains the N Hankel singular values
              of the extended system ordered decreasingly (see METHOD).

Tolerances

      TOL1    DOUBLE PRECISION
              If ORDSEL = 'A', TOL1 contains the tolerance for
              determining the order of the reduced extended system.
              For model reduction, the recommended value is
              TOL1 = c*HNORM(Ge), where c is a constant in the
              interval [0.00001,0.001], and HNORM(Ge) is the
              Hankel norm of the extended system (computed in HSV(1)).
              The value TOL1 = N*EPS*HNORM(Ge) is used by default if
              TOL1 <= 0 on entry, where EPS is the machine precision
              (see LAPACK Library Routine DLAMCH).
              If ORDSEL = 'F', the value of TOL1 is ignored.

      TOL2    DOUBLE PRECISION
              The tolerance for determining the order of a minimal
              realization of the coprime factorization controller
              (see METHOD). The recommended value is
              TOL2 = N*EPS*HNORM(Ge) (see METHOD).
              This value is used by default if TOL2 <= 0 on entry.
              If TOL2 > 0 and ORDSEL = 'A', then TOL2 <= TOL1.

Workspace

      IWORK   INTEGER array, dimension (LIWORK)
              LIWORK = 0,         if ORDSEL = 'F' and NCR = N.
                                                  Otherwise,
              LIWORK = MAX(PM,M), if JOBCF = 'L',
              LIWORK = MAX(PM,P), if JOBCF = 'R', where
              PM = 0,             if JOBMR = 'B',
              PM = N,             if JOBMR = 'F',
              PM = MAX(1,2*N),    if JOBMR = 'S' or '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 >= P*N, if ORDSEL = 'F' and NCR = N. Otherwise,
              LDWORK >= (N+M)*(M+P) + MAX(LWR,4*M), if JOBCF = 'L',
              LDWORK >= (N+P)*(M+P) + MAX(LWR,4*P), if JOBCF = 'R',
              where LWR = MAX(1,N*(2*N+MAX(N,M+P)+5)+N*(N+1)/2).
              For optimum performance LDWORK should be larger.

Warning Indicator

      IWARN   INTEGER
              = 0:  no warning;
              = 1:  with ORDSEL = 'F', the selected order NCR is
                    greater than the order of a minimal
                    realization of the controller.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              = 1:  the reduction of A+G*C to a real Schur form
                    failed;
              = 2:  the matrix A+G*C is not stable (if DICO = 'C'),
                    or not convergent (if DICO = 'D');
              = 3:  the computation of Hankel singular values failed;
              = 4:  the reduction of A+B*F to a real Schur form
                    failed;
              = 5:  the matrix A+B*F is not stable (if DICO = 'C'),
                    or not convergent (if DICO = 'D').

Method

      Let be the linear system

           d[x(t)] = Ax(t) + Bu(t)
           y(t)    = Cx(t) + Du(t),                             (1)

      where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
      for a discrete-time system, and let Go(d) be the open-loop
      transfer-function matrix
                            -1
           Go(d) = C*(d*I-A) *B + D .

      Let F and G be the state feedback and observer gain matrices,
      respectively, chosen so that A+B*F and A+G*C are stable matrices.
      The controller has a transfer-function matrix K(d) given by
                                         -1
           K(d) = F*(d*I-A-B*F-G*C-G*D*F) *G .

      The closed-loop transfer-function matrix is given by
                                      -1
           Gcl(d) = Go(d)(I+K(d)Go(d)) .

      K(d) can be expressed as a left coprime factorization (LCF),
                           -1
           K(d) = M_left(d) *N_left(d) ,

      or as a right coprime factorization (RCF),
                                       -1
           K(d) = N_right(d)*M_right(d) ,

      where M_left(d), N_left(d), N_right(d), and M_right(d) are
      stable transfer-function matrices.

      The subroutine SB16BD determines the matrices of a reduced
      controller

           d[z(t)] = Ac*z(t) + Bc*y(t)
           u(t)    = Cc*z(t) + Dc*y(t),                           (2)

      with the transfer-function matrix Kr as follows:

      (1) If JOBCF = 'L', the extended system
          Ge(d)  = [ N_left(d) M_left(d) ] is reduced to
          Ger(d) = [ N_leftr(d) M_leftr(d) ] by using either the
          B&T or SPA methods. The reduced order controller Kr(d)
          is computed as
                            -1
          Kr(d) = M_leftr(d) *N_leftr(d) ;

      (2) If JOBCF = 'R', the extended system
          Ge(d) = [ N_right(d) ] is reduced to
                  [ M_right(d) ]
          Ger(d) = [ N_rightr(d) ] by using either the
                   [ M_rightr(d) ]
          B&T or SPA methods. The reduced order controller Kr(d)
          is computed as
                                          -1
          Kr(d) = N_rightr(d)* M_rightr(d) .

      If ORDSEL = 'A', the order of the controller is determined by
      computing the number of Hankel singular values greater than
      the given tolerance TOL1. The Hankel singular values are
      the square roots of the eigenvalues of the product of
      the controllability and observability Grammians of the
      extended system Ge.

      If JOBMR = 'B', the square-root B&T method of [1] is used.

      If JOBMR = 'F', the balancing-free square-root version of the
      B&T method [1] is used.

      If JOBMR = 'S', the square-root version of the SPA method [2,3]
      is used.

      If JOBMR = 'P', the balancing-free square-root version of the
      SPA method [2,3] is used.

References

      [1] Tombs, M.S. and Postlethwaite, I.
          Truncated balanced realization of stable, non-minimal
          state-space systems.
          Int. J. Control, Vol. 46, pp. 1319-1330, 1987.

      [2] Varga, A.
          Efficient minimal realization procedure based on balancing.
          Proc. of IMACS/IFAC Symp. MCTS, Lille, France, May 1991,
          A. El Moudui, P. Borne, S. G. Tzafestas (Eds.), Vol. 2,
          pp. 42-46, 1991.

      [3] 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.

      [4] Liu, Y., Anderson, B.D.O. and Ly, O.L.
          Coprime factorization controller reduction with Bezout
          identity induced frequency weighting.
          Automatica, vol. 26, pp. 233-249, 1990.

Numerical Aspects

      The implemented methods rely on accuracy enhancing square-root or
      balancing-free square-root techniques.
                                          3
      The algorithms require less than 30N  floating point operations.

Further Comments

      None

Example

Program Text

    *     SB16BD EXAMPLE PROGRAM TEXT
    *
    *     .. Parameters ..
          INTEGER          NIN, NOUT
          PARAMETER        ( NIN = 5, NOUT = 6 )
          INTEGER          NMAX, MMAX, PMAX
          PARAMETER        ( NMAX = 20, MMAX = 20, PMAX = 20 )
          INTEGER          LDA, LDB, LDC, LDD, LDDC, LDF, LDG
          PARAMETER        ( LDA = NMAX, LDB  = NMAX, LDC = PMAX,
         $                   LDD = PMAX, LDDC = MMAX, LDF = MMAX, LDG = NMAX
         $                 )
          INTEGER          LDWORK, LIWORK, MAXMP, MPMAX
          PARAMETER        ( LIWORK = 2*NMAX, MAXMP = MAX( MMAX, PMAX ),
         $                   MPMAX  = MMAX + PMAX )
          PARAMETER        ( LDWORK = ( NMAX + MAXMP )*MPMAX +
         $                            MAX ( NMAX*( 2*NMAX +
         $                                         MAX( NMAX, MPMAX ) + 5 )
         $                                      + ( NMAX*( NMAX + 1 ) )/2,
         $                                  4*MAXMP ) )
          CHARACTER        DICO, EQUIL, JOBCF, JOBD, JOBMR, ORDSEL
          INTEGER          I, INFO, IWARN, J, M, N, NCR, P
          DOUBLE PRECISION TOL1, TOL2
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
         $                 D(LDD,MMAX), DC(LDDC,PMAX), DWORK(LDWORK),
         $                 F(LDF,NMAX), G(LDG,PMAX), HSV(NMAX)
          INTEGER          IWORK(LIWORK)
    *     .. External Subroutines ..
          EXTERNAL         SB16BD
    *     .. 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, NCR, TOL1, TOL2,
         $                      DICO, JOBD, JOBMR, JOBCF, EQUIL, ORDSEL
          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 )
                   READ ( NIN, FMT = * ) ( ( F(I,J), J = 1,N ), I = 1,M )
                   READ ( NIN, FMT = * ) ( ( G(I,J), J = 1,P ), I = 1,N )
    *              Find a reduced ssr for (A,B,C,D).
                   CALL SB16BD( DICO, JOBD, JOBMR, JOBCF, EQUIL, ORDSEL, N,
         $                      M, P, NCR, A, LDA, B, LDB, C, LDC, D, LDD,
         $                      F, LDF, G, LDG, DC, LDDC, HSV, TOL1, TOL2,
         $                      IWORK, DWORK, LDWORK, IWARN, INFO )
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      WRITE ( NOUT, FMT = 99997 ) NCR
                      WRITE ( NOUT, FMT = 99987 )
                      WRITE ( NOUT, FMT = 99995 ) ( HSV(J), J = 1,N )
                      IF( NCR.GT.0 ) WRITE ( NOUT, FMT = 99996 )
                      DO 20 I = 1, NCR
                         WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NCR )
       20             CONTINUE
                      IF( NCR.GT.0 ) WRITE ( NOUT, FMT = 99993 )
                      DO 40 I = 1, NCR
                         WRITE ( NOUT, FMT = 99995 ) ( G(I,J), J = 1,P )
       40             CONTINUE
                      IF( NCR.GT.0 ) WRITE ( NOUT, FMT = 99992 )
                      DO 60 I = 1, M
                         WRITE ( NOUT, FMT = 99995 ) ( F(I,J), J = 1,NCR )
       60             CONTINUE
                      WRITE ( NOUT, FMT = 99991 )
                      DO 80 I = 1, P
                         WRITE ( NOUT, FMT = 99995 ) ( DC(I,J), J = 1,M )
       80             CONTINUE
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' SB16BD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from SB16BD = ',I2)
    99997 FORMAT (' The order of reduced controller = ',I2)
    99996 FORMAT (/' The reduced controller state dynamics matrix Ac is ')
    99995 FORMAT (20(1X,F8.4))
    99993 FORMAT (/' The reduced controller input/state matrix Bc is ')
    99992 FORMAT (/' The reduced controller state/output matrix Cc is ')
    99991 FORMAT (/' The reduced controller input/output matrix Dc 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)
    99987 FORMAT (/' The Hankel singular values of extended system are:')
          END

Program Data

     SB16BD EXAMPLE PROGRAM DATA (Continuous system)
      8  1  1   4   0.1E0  0.0    C  D  F  L  S   F
             0    1.0000         0         0         0         0         0        0
             0         0         0         0         0         0         0        0
             0         0   -0.0150    0.7650         0         0         0        0
             0         0   -0.7650   -0.0150         0         0         0        0
             0         0         0         0   -0.0280    1.4100         0        0
             0         0         0         0   -1.4100   -0.0280         0        0
             0         0         0         0         0         0   -0.0400    1.850
             0         0         0         0         0         0   -1.8500   -0.040
        0.0260
       -0.2510
        0.0330
       -0.8860
       -4.0170
        0.1450
        3.6040
        0.2800
      -.996 -.105 0.261 .009 -.001 -.043 0.002 -0.026
      0.0
    4.4721e-002  6.6105e-001  4.6986e-003  3.6014e-001  1.0325e-001 -3.7541e-002 -4.2685e-002  3.2873e-002
      4.1089e-001
      8.6846e-002
      3.8523e-004
     -3.6194e-003
     -8.8037e-003
      8.4205e-003
      1.2349e-003
      4.2632e-003

Program Results

     SB16BD EXAMPLE PROGRAM RESULTS

     The order of reduced controller =  4

     The Hankel singular values of extended system are:
       4.9078   4.8745   3.8455   3.7811   1.2289   1.1785   0.5176   0.1148

     The reduced controller state dynamics matrix Ac is
       0.5946  -0.7336   0.1914  -0.3368
       0.5960  -0.0184  -0.1088   0.0207
       1.2253   0.2043   0.1009  -1.4948
      -0.0330  -0.0243   1.3440   0.0035

     The reduced controller input/state matrix Bc is
       0.0015
      -0.0202
       0.0159
      -0.0544

     The reduced controller state/output matrix Cc is
       0.3534   0.0274   0.0337  -0.0320

     The reduced controller input/output matrix Dc is
       0.0000

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