AB09HD

Stochastic balancing based model reduction

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

Purpose

  To compute a reduced order model (Ar,Br,Cr,Dr) for an original
  state-space representation (A,B,C,D) by using the stochastic 
  balancing approach in conjunction with the square-root or 
  the balancing-free square-root Balance & Truncate (B&T) 
  or Singular Perturbation Approximation (SPA) model reduction
  methods for the ALPHA-stable part of the system.

Specification
      SUBROUTINE AB09HD( DICO, JOB, EQUIL, ORDSEL, N, M, P, NR, ALPHA, 
     $                   BETA, A, LDA, B, LDB, C, LDC, D, LDD, NS, HSV, 
     $                   TOL1, TOL2, IWORK, DWORK, LDWORK, BWORK, IWARN,
     $                   INFO )
C     .. Scalar Arguments ..
      CHARACTER         DICO, EQUIL, JOB, ORDSEL
      INTEGER           INFO, IWARN, LDA, LDB, LDC, LDD, LDWORK, 
     $                  M, N, NR, NS, P
      DOUBLE PRECISION  ALPHA, BETA, TOL1, TOL2
C     .. Array Arguments ..
      INTEGER           IWORK(*)
      DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*), 
     $                  DWORK(*), HSV(*)
      LOGICAL           BWORK(*)

Arguments

Mode Parameters

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

  JOB     CHARACTER*1  
          Specifies the model reduction approach to be used 
          as follows:
          = 'B':  use the square-root Balance & Truncate method;
          = 'F':  use the balancing-free square-root 
                  Balance & Truncate method;
          = 'S':  use the square-root Singular Perturbation 
                  Approximation method;
          = 'P':  use the balancing-free square-root 
                  Singular Perturbation Approximation method.

  EQUIL   CHARACTER*1
          Specifies whether the user wishes to preliminarily
          equilibrate the triplet (A,B,C) as follows:
          = 'S':  perform equilibration (scaling);
          = 'N':  do not perform equilibration.

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

Input/Output Parameters
  N       (input) INTEGER
          The order of the original state-space representation,
          i.e., the order of the matrix A.  N >= 0.

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

  P       (input) INTEGER
          The number of system outputs.  P >= 0.
          P <= M if BETA = 0.

  NR      (input/output) INTEGER
          On entry with ORDSEL = 'F', NR is the desired order of the
          resulting reduced order system.  0 <= NR <= N.
          On exit, if INFO = 0, NR is the order of the resulting 
          reduced order model. For a system with NU ALPHA-unstable 
          eigenvalues and NS ALPHA-stable eigenvalues (NU+NS = N),
          NR is set as follows: if ORDSEL = 'F', NR is equal to 
          NU+MIN(MAX(0,NR-NU),NMIN), where NR is the desired order 
          on entry, and NMIN is the order of a minimal realization
          of the ALPHA-stable part of the given system; NMIN is
          determined as the number of Hankel singular values greater
          than NS*EPS, where EPS is the machine precision 
          (see LAPACK Library Routine DLAMCH);
          if ORDSEL = 'A', NR is the sum of NU and the number of
          Hankel singular values greater than MAX(TOL1,NS*EPS);
          NR can be further reduced to ensure that 
          HSV(NR-NU) > HSV(NR+1-NU).
          
  ALPHA   (input) DOUBLE PRECISION
          Specifies the ALPHA-stability boundary for the eigenvalues
          of the state dynamics matrix A. For a continuous-time
          system (DICO = 'C'), ALPHA <= 0 is the boundary value for
          the real parts of eigenvalues, while for a discrete-time
          system (DICO = 'D'), 0 <= ALPHA <= 1 represents the
          boundary value for the moduli of eigenvalues.
          The ALPHA-stability domain does not include the boundary.

  BETA    (input) DOUBLE PRECISION
          BETA > 0 specifies the absolute/relative error weighting
          parameter. A large positive value of BETA favours the
          minimization of the absolute approximation error, while a
          small value of BETA is appropriate for the minimization
          of the relative error.
          BETA = 0 means a pure relative error method and can be
          used only if rank(D) = P.

  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, if INFO = 0, the leading NR-by-NR part of this 
          array contains the state dynamics matrix Ar of the reduced
          order system.
          The resulting A has a block-diagonal form with two blocks.
          For a system with NU ALPHA-unstable eigenvalues and 
          NS ALPHA-stable eigenvalues (NU+NS = N), the leading
          NU-by-NU block contains the unreduced part of A 
          corresponding to ALPHA-unstable eigenvalues in an
          upper real Schur form. 
          The trailing (NR+NS-N)-by-(NR+NS-N) block contains 
          the reduced part of A corresponding to ALPHA-stable 
          eigenvalues.

  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 original input/state matrix B.
          On exit, if INFO = 0, the leading NR-by-M part of this
          array contains the input/state matrix Br of the reduced
          order system.

  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 original state/output matrix C.
          On exit, if INFO = 0, the leading P-by-NR part of this
          array contains the state/output matrix Cr of the reduced
          order system.

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

  D       (input/output) DOUBLE PRECISION array, dimension (LDD,M)
          On entry, the leading P-by-M part of this array must 
          contain the original input/output matrix D.
          On exit, if INFO = 0, the leading P-by-M part of this
          array contains the input/output matrix Dr of the reduced
          order system.

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

  NS      (output) INTEGER
          The dimension of the ALPHA-stable subsystem.

  HSV     (output) DOUBLE PRECISION array, dimension (N)
          If INFO = 0, the leading NS elements of HSV contain the
          Hankel singular values of the phase system corresponding 
          to the ALPHA-stable part of the original system.
          The Hankel singular values are ordered decreasingly. 

Tolerances
  TOL1    DOUBLE PRECISION
          If ORDSEL = 'A', TOL1 contains the tolerance for 
          determining the order of reduced system.
          For model reduction, the recommended value of TOL1 lies 
          in the interval [0.00001,0.001].
          If TOL1 <= 0 on entry, the used default value is 
          TOL1 = NS*EPS, where NS is the number of 
          ALPHA-stable eigenvalues of A and EPS is the machine
          precision (see LAPACK Library Routine DLAMCH).
          If ORDSEL = 'F', the value of TOL1 is ignored.
          TOL1 < 1.

  TOL2    DOUBLE PRECISION
          The tolerance for determining the order of a minimal
          realization of the phase system (see METHOD) corresponding
          to the ALPHA-stable part of the given system. 
          The recommended value is TOL2 = NS*EPS. 
          This value is used by default if TOL2 <= 0 on entry.
          If TOL2 > 0 and ORDSEL = 'A', then TOL2 <= TOL1.
          TOL2 < 1.

Workspace
  IWORK   INTEGER array, dimension MAX(1,2*N)
          On exit with INFO = 0, IWORK(1) contains the order of the
          minimal realization of the system.

  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          On exit, if INFO = 0, DWORK(1) returns the optimal value
          of LDWORK and DWORK(2) contains RCOND, the reciprocal 
          condition number of the U11 matrix from the expression 
          used to compute the solution X = U21*inv(U11) of the 
          Riccati equation for spectral factorization. 
          A small value RCOND indicates possible ill-conditioning 
          of the respective Riccati equation.

  LDWORK  INTEGER
          The length of the array DWORK.
          LDWORK >= 2*N*N + MB*(N+P) + MAX( 2, N*(MAX(N,MB,P)+5), 
                                 2*N*P+MAX(P*(MB+2),10*N*(N+1) ) ),
          where MB = M if BETA = 0 and MB = M+P if BETA > 0.
          For optimum performance LDWORK should be larger.

  BWORK   LOGICAL array, dimension 2*N

Warning Indicator
  IWARN   INTEGER
          = 0:  no warning;
          = 1:  with ORDSEL = 'F', the selected order NR is greater
                than NSMIN, the sum of the order of the
                ALPHA-unstable part and the order of a minimal
                realization of the ALPHA-stable part of the given
                system; in this case, the resulting NR is set equal
                to NSMIN;
          = 2:  with ORDSEL = 'F', the selected order NR corresponds
                to repeated singular values for the ALPHA-stable 
                part, which are neither all included nor all 
                excluded from the reduced model; in this case, the
                resulting NR is automatically decreased to exclude
                all repeated singular values; 
          = 3:  with ORDSEL = 'F', the selected order NR is less
                than the order of the ALPHA-unstable part of the 
                given system; in this case NR is set equal to the
                order of the ALPHA-unstable part.

Error Indicator
  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value;
          = 1:  the computation of the ordered real Schur form of A
                failed;
          = 2:  the reduction of the Hamiltonian matrix to real 
                Schur form failed;
          = 3:  the reordering of the real Schur form of the 
                Hamiltonian matrix failed;
          = 4:  the Hamiltonian matrix has less than N stable 
                eigenvalues;
          = 5:  the coefficient matrix U11 in the linear system 
                X*U11 = U21 to determine X is singular to working 
                precision;
          = 6:  BETA = 0 and D has not a maximal row rank;
          = 7:  the computation of Hankel singular values failed;
          = 8:  the separation of the ALPHA-stable/unstable diagonal
                blocks failed because of very close eigenvalues;
          = 9:  the resulting order of reduced stable part is less 
                than the number of unstable zeros of the stable 
                part.
Method
  Let be the following 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. The subroutine AB09HD determines for
  the given system (1), the matrices of a reduced order system

       d[z(t)] = Ar*z(t) + Br*u(t)
       yr(t)   = Cr*z(t) + Dr*u(t),                  (2)

  such that  

       INFNORM[inv(conj(W))*(G-Gr)] <= 
                    (1+HSV(NR+NS-N+1)) / (1-HSV(NR+NS-N+1)) + ...
                    + (1+HSV(NS)) / (1-HSV(NS)) - 1,

  where G and Gr are transfer-function matrices of the systems
  (A,B,C,D) and (Ar,Br,Cr,Dr), respectively, W is the right, minimum
  phase spectral factor satisfying
         
      G1*conj(G1) = conj(W)* W,                      (3)

  G1 is the NS-order ALPHA-stable part of G, and INFNORM(G) is the 
  infinity-norm of G. HSV(1), ... , HSV(NS) are the Hankel-singular 
  values of the stable part of the phase system (Ap,Bp,Cp) 
  with the transfer-function matrix 

       P = inv(conj(W))*G1.

  If BETA > 0, then the model reduction is performed on [G BETA*I]
  instead of G. This is the recommended approach to be used when D
  has not a maximal row rank or when a certain balance between
  relative and absolute approximation errors is desired. For
  increasingly large values of BETA, the obtained reduced system
  assymptotically approaches that computed by using the
  Balance & Truncate or Singular Perturbation Approximation methods.

  Note: conj(G)  denotes either G'(-s) for a continuous-time system
        or G'(1/z) for a discrete-time system. 
        inv(G) is the inverse of G.

  The following procedure is used to reduce a given G:

  1) Decompose additively G as

       G = G1 + G2,

     such that G1 = (As,Bs,Cs,D) has only ALPHA-stable poles and 
     G2 = (Au,Bu,Cu) has only ALPHA-unstable poles.

  2) Determine G1r, a reduced order approximation of the 
     ALPHA-stable part G1 using the balancing stochastic method
     in conjunction with either the B&T [1,2] or SPA methods [3].

  3) Assemble the reduced model Gr as

        Gr = G1r + G2.

  Note: The employed stochastic truncation algorithm [2,3] has the  
  property that right half plane zeros of G1 remain as right half 
  plane zeros of G1r. Thus, the order can not be chosen smaller than
  the sum of the number of unstable poles of G and the number of 
  unstable zeros of G1.  

  The reduction of the ALPHA-stable part G1 is done as follows.

  If JOB = 'B', the square-root stochastic Balance & Truncate 
  method of [1] is used. 
  For an ALPHA-stable continuous-time system (DICO = 'C'), 
  the resulting reduced model is stochastically balanced. 

  If JOB = 'F', the balancing-free square-root version of the
  stochastic Balance & Truncate method [1] is used to reduce 
  the ALPHA-stable part G1. 

  If JOB = 'S', the stochastic balancing method is used to reduce 
  the ALPHA-stable part G1, in conjunction with the square-root 
  version of the Singular Perturbation Approximation method [3,4]. 

  If JOB = 'P', the stochastic balancing method is used to reduce 
  the ALPHA-stable part G1, in conjunction with the balancing-free
  square-root version of the Singular Perturbation Approximation 
  method [3,4]. 

References
  [1] Varga A. and Fasol K.H.
      A new square-root balancing-free stochastic truncation model
      reduction algorithm.
      Proc. 12th IFAC World Congress, Sydney, 1993.

  [2] Safonov M. G. and Chiang R. Y.
      Model reduction for robust control: a Schur relative error
      method.
      Int. J. Adapt. Contr. Sign. Proc., vol. 2, pp. 259-272, 1988.

  [3] Green M. and Anderson B. D. O.
      Generalized balanced stochastic truncation.
      Proc. 29-th CDC, Honolulu, Hawaii, pp. 476-481, 1990.

  [4] Varga A.
      Balancing-free square-root algorithm for computing
      singular perturbation approximations.
      Proc. 30-th IEEE CDC,  Brighton, Dec. 11-13, 1991, 
      Vol. 2, pp. 1062-1065.

Numerical Aspects
                            
  The implemented methods rely on accuracy enhancing square-root or
  balancing-free square-root techniques. The effectiveness of the 
  accuracy enhancing technique depends on the accuracy of the 
  solution of a Riccati equation. An ill-conditioned Riccati 
  solution typically results when [D BETA*I] is nearly 
  rank deficient.
                                   3
  The algorithm requires about 100N  floating point operations.

Further Comments
  None
Example

Program Text

*     AB09HD EXAMPLE PROGRAM TEXT
*     RELEASE 4.5, Copyright (c) 2002-2005 NICONET Int. Society.
*
*     .. 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
      PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX, 
     $                 LDD = PMAX )
      INTEGER          LBWORK, LIWORK
      PARAMETER        ( LBWORK = 2*NMAX, LIWORK = 2*NMAX )
      INTEGER          LDWORK, MBMAX
      PARAMETER        ( MBMAX = MMAX + PMAX )
      PARAMETER        ( LDWORK = 2*NMAX*NMAX + MBMAX*(NMAX+PMAX) +
     $                      MAX( NMAX*(MAX( NMAX, MMAX, PMAX) + 5),
     $                     2*NMAX*PMAX + MAX( PMAX*(MBMAX+2), 
     $                                        10*NMAX*(NMAX+1) ) ) )
*     .. Local Scalars ..
      DOUBLE PRECISION ALPHA, BETA, TOL1, TOL2
      INTEGER          I, INFO, IWARN, J, M, N, NR, NS, P
      CHARACTER*1      DICO, EQUIL, JOB, ORDSEL
*     .. Local Arrays ..
      DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
     $                 D(LDD,MMAX), DWORK(LDWORK), HSV(NMAX)
      LOGICAL          BWORK(LBWORK)
      INTEGER          IWORK(LIWORK)
*     .. External Subroutines ..
      EXTERNAL         AB09HD
*     .. 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, NR, ALPHA, BETA, TOL1, TOL2,
     $                      DICO, JOB, 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 )
*              Find a reduced ssr for (A,B,C,D).
               CALL AB09HD( DICO, JOB, EQUIL, ORDSEL, N, M, P, NR, 
     $                      ALPHA, BETA, A, LDA, B, LDB, C, LDC, D, LDD,
     $                      NS, HSV, TOL1, TOL2, IWORK, DWORK, LDWORK, 
     $                      BWORK, IWARN, INFO )
*
               IF ( INFO.NE.0 ) THEN
                  WRITE ( NOUT, FMT = 99998 ) INFO
               ELSE
                  WRITE ( NOUT, FMT = 99997 ) NR
                  WRITE ( NOUT, FMT = 99987 )
                  WRITE ( NOUT, FMT = 99995 ) ( HSV(J), J = 1,NS )
                  IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99996 )
                  DO 20 I = 1, NR
                     WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR )
   20             CONTINUE
                  IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99993 )
                  DO 40 I = 1, NR
                     WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
   40             CONTINUE
                  IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99992 )
                  DO 60 I = 1, P
                     WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR )
   60             CONTINUE
                  WRITE ( NOUT, FMT = 99991 )
                  DO 70 I = 1, P
                     WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1,M )
   70             CONTINUE
               END IF
            END IF
         END IF
      END IF
      STOP
*
99999 FORMAT (' AB09HD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB09HD = ',I2)
99997 FORMAT (' The order of reduced model = ',I2)
99996 FORMAT (/' The reduced state dynamics matrix Ar is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The reduced input/state matrix Br is ')
99992 FORMAT (/' The reduced state/output matrix Cr is ')
99991 FORMAT (/' The reduced input/output matrix Dr 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 stochastic Hankel singular values of ALPHA-stable'
     $        ,' part are')
      END
Program Data
 AB09HD EXAMPLE PROGRAM DATA (Continuous system)
  7  2   3   0   0.0   1.0  0.1E0  0.0    C     F     N     A
 -0.04165  0.0000  4.9200  -4.9200  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 -0.5450  0.0000  0.0000
  0.0000   0.0000  0.0000   4.9200 -0.04165 0.0000  4.9200
  0.0000   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
  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
 AB09HD EXAMPLE PROGRAM RESULTS

 The order of reduced model =  5

 The stochastic Hankel singular values of ALPHA-stable part are
   0.8803   0.8506   0.8038   0.4494   0.3973   0.0214   0.0209

 The reduced state dynamics matrix Ar is 
   1.2729   0.0000   6.5947   0.0000  -3.4229
   0.0000   0.8169   0.0000   2.4821   0.0000
  -2.9889   0.0000  -2.9028   0.0000  -0.3692
   0.0000  -3.3921   0.0000  -3.1126   0.0000
  -1.4767   0.0000  -2.0339   0.0000  -0.6107

 The reduced input/state matrix Br is 
   0.1331  -0.1331
  -0.0862  -0.0862
  -2.6777   2.6777
  -3.5767  -3.5767
  -2.3033   2.3033

 The reduced state/output matrix Cr is 
  -0.6907  -0.6882   0.0779   0.0958  -0.0038
   0.0676   0.0000   0.6532   0.0000  -0.7522
   0.6907  -0.6882  -0.0779   0.0958   0.0038

 The reduced input/output matrix Dr is 
   0.0000   0.0000
   0.0000   0.0000
   0.0000   0.0000

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