## AB09HX

### Stochastic balancing model reduction of stable systems

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

Purpose

      To compute a reduced order model (Ar,Br,Cr,Dr) for an original
      stable 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.
      The state dynamics matrix A of the original system is an upper
      quasi-triangular matrix in real Schur canonical form and D must be
      full row rank.

      For the B&T approach, the matrices of the reduced order system
      are computed using the truncation formulas:

           Ar = TI * A * T ,  Br = TI * B ,  Cr = C * T .     (1)

      For the SPA approach, the matrices of a minimal realization
      (Am,Bm,Cm) are computed using the truncation formulas:

           Am = TI * A * T ,  Bm = TI * B ,  Cm = C * T .     (2)

      Am, Bm, Cm and D serve further for computing the SPA of the given
      system.

Specification

          SUBROUTINE AB09HX( DICO, JOB, ORDSEL, N, M, P, NR, A, LDA, B, LDB,
         $                   C, LDC, D, LDD, HSV, T, LDT, TI, LDTI, TOL1,
         $                   TOL2, IWORK, DWORK, LDWORK, BWORK, IWARN,
         $                   INFO )
    C     .. Scalar Arguments ..
          CHARACTER         DICO, JOB, ORDSEL
          INTEGER           INFO, IWARN, LDA, LDB, LDC, LDD, LDT, LDTI,
         $                  LDWORK, M, N, NR, P
          DOUBLE PRECISION  TOL1, TOL2
    C     .. Array Arguments ..
          INTEGER           IWORK(*)
          DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
         $                  DWORK(*), HSV(*), T(LDT,*), TI(LDTI,*)
          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.

      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.  M >= P >= 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. NR is set as follows:
              if ORDSEL = 'F', NR is equal to MIN(NR,NMIN), where NR
              is the desired order on entry and NMIN is the order of a
              minimal realization of the given system; NMIN is
              determined as the number of Hankel singular values greater
              than N*EPS, where EPS is the machine precision
              (see LAPACK Library Routine DLAMCH);
              if ORDSEL = 'A', NR is equal to the number of Hankel
              singular values greater than MAX(TOL1,N*EPS).

      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 in a real Schur
              canonical form.
              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.

      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).

      HSV     (output) DOUBLE PRECISION array, dimension (N)
              If INFO = 0, it contains the Hankel singular values,
              ordered decreasingly, of the phase system. All singular
              values are less than or equal to 1.

      T       (output) DOUBLE PRECISION array, dimension (LDT,N)
              If INFO = 0 and NR > 0, the leading N-by-NR part of this
              array contains the right truncation matrix T in (1), for
              the B&T approach, or in (2), for the SPA approach.

      LDT     INTEGER
              The leading dimension of array T.  LDT >= MAX(1,N).

      TI      (output) DOUBLE PRECISION array, dimension (LDTI,N)
              If INFO = 0 and NR > 0, the leading NR-by-N part of this
              array contains the left truncation matrix TI in (1), for
              the B&T approach, or in (2), for the SPA approach.

      LDTI    INTEGER
              The leading dimension of array TI.  LDTI >= MAX(1,N).

Tolerances

      TOL1    DOUBLE PRECISION
              If ORDSEL = 'A', TOL1 contains the tolerance for
              determining the order of reduced system.
              For model reduction, the recommended value lies in the
              interval [0.00001,0.001].
              If TOL1 <= 0 on entry, the used default value is
              TOL1 = N*EPS, 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 phase system (see METHOD) corresponding
              to the given system.
              The recommended value is TOL2 = N*EPS.
              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 (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 >= MAX( 2, N*(MAX(N,M,P)+5),
                             2*N*P+MAX(P*(M+2),10*N*(N+1) ) ).
              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 the order of a minimal realization of the
                    given system. In this case, the resulting NR is
                    set automatically to a value corresponding to the
                    order of a minimal realization of the system.

Error Indicator

      INFO    INTEGER
              = 0:  successful exit;
              < 0:  if INFO = -i, the i-th argument had an illegal
                    value;
              = 1:  the state matrix A is not stable (if DICO = 'C')
                    or not convergent (if DICO = 'D'), or it is not in
                    a real Schur form;
              = 2:  the reduction of 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, used to determine X, is singular to
                    working precision;
              = 6:  the feedthrough matrix D has not a full row rank P;
              = 7:  the computation of Hankel singular values failed.

Method

      Let be the stable linear system

           d[x(t)] = Ax(t) + Bu(t)
           y(t)    = Cx(t) + Du(t),                             (3)

      where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
      for a discrete-time system. The subroutine AB09HX determines for
      the given system (3), the matrices of a reduced NR-rder system

           d[z(t)] = Ar*z(t) + Br*u(t)
           yr(t)   = Cr*z(t) + Dr*u(t),                         (4)

      such that

            HSV(NR) <= INFNORM(G-Gr) <= 2*[HSV(NR+1) + ... + HSV(N)],

      where G and Gr are transfer-function matrices of the systems
      (A,B,C,D) and (Ar,Br,Cr,Dr), respectively, and INFNORM(G) is the
      infinity-norm of G.

      If JOB = 'B', the square-root stochastic Balance & Truncate
      method of [1] is used and the resulting model is balanced.

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

      If JOB = 'S', the stochastic balancing method, in conjunction
      with the square-root version of the Singular Perturbation
      Approximation method [2,3] is used.

      If JOB = 'P', the stochastic balancing method, in conjunction
      with the balancing-free square-root version of the Singular
      Perturbation Approximation method [2,3] is used.

      By setting TOL1 = TOL2, the routine can be also used to compute
      Balance & Truncate approximations.

References

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

      [2] Liu Y. and Anderson B.D.O.
          Singular Perturbation Approximation of balanced systems.
          Int. J. Control, Vol. 50, pp. 1379-1405, 1989.

      [3] 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 method relies on accuracy enhancing square-root
      or balancing-free square-root methods. The effectiveness of the
      accuracy enhancing technique depends on the accuracy of the
      solution of a Riccati equation. Ill-conditioned Riccati solution
      typically results when D is nearly rank deficient.
                                       3
      The algorithm requires about 100N  floating point operations.

Further Comments

      None

Example

Program Text

      None

Program Data

      None

Program Results

      None

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