## AB13AD

### Hankel-norm of a stable projection

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

Purpose

      To compute the Hankel-norm of the ALPHA-stable projection of the
      transfer-function matrix G of the state-space system (A,B,C).

Specification

          DOUBLE PRECISION FUNCTION AB13AD( DICO, EQUIL, N, M, P, ALPHA, A,
         $                                  LDA, B, LDB, C, LDC, NS, HSV,
         $                                  DWORK, LDWORK, INFO )
    C     .. Scalar Arguments ..
          CHARACTER         DICO, EQUIL
          INTEGER           INFO, LDA, LDB, LDC, LDWORK, M, N, NS, P
          DOUBLE PRECISION  ALPHA
    C     .. Array Arguments ..
          DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), HSV(*)

Function Value

      AB13AD  DOUBLE PRECISION
              The Hankel-norm of the ALPHA-stable projection of G
              (if INFO = 0).

Arguments

Mode Parameters

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

      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.

Input/Output Parameters

      N       (input) INTEGER
              The order of the 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.

      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
              (see the Note below).

      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 N-by-N part of this
              array contains the state dynamics matrix A in a block
              diagonal real Schur form with its eigenvalues reordered
              and separated. The resulting A has two diagonal blocks.
              The leading NS-by-NS part of A has eigenvalues in the
              ALPHA-stability domain and the trailing (N-NS) x (N-NS)
              part has eigenvalues outside the ALPHA-stability domain.
              Note: The ALPHA-stability domain is defined either
                    as the open half complex plane left to ALPHA,
                    for a continous-time system (DICO = 'C'), or the
                    interior of the ALPHA-radius circle centered in the
                    origin, for a discrete-time system (DICO = 'D').

      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 N-by-M part of this
              array contains the input/state matrix B of the transformed
              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-N part of this
              array contains the state/output matrix C of the
              transformed system.

      LDC     INTEGER
              The leading dimension of array C.  LDC >= 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 ALPHA-stable part of the
              original system ordered decreasingly.
              HSV(1) is the Hankel norm of the ALPHA-stable subsystem.

Workspace

      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,N*(MAX(N,M,P)+5)+N*(N+1)/2).
              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:  the computation of the ordered real Schur form of A
                    failed;
              = 2:  the separation of the ALPHA-stable/unstable diagonal
                    blocks failed because of very close eigenvalues;
              = 3:  the computed ALPHA-stable part is just stable,
                    having stable eigenvalues very near to the imaginary
                    axis (if DICO = 'C') or to the unit circle
                    (if DICO = 'D');
              = 4:  the computation of Hankel singular values failed.

Method

      Let be the following linear system

           d[x(t)] = Ax(t) + Bu(t)
           y(t)    = Cx(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 G be the corresponding
      transfer-function matrix. The following procedure is used to
      compute the Hankel-norm of the ALPHA-stable projection of G:

      1) Decompose additively G as

           G = G1 + G2

         such that G1 = (As,Bs,Cs) has only ALPHA-stable poles and
         G2 = (Au,Bu,Cu) has only ALPHA-unstable poles.
         For the computation of the additive decomposition, the
         algorithm presented in [1] is used.

      2) Compute the Hankel-norm of ALPHA-stable projection G1 as the
         the maximum Hankel singular value of the system (As,Bs,Cs).
         The computation of the Hankel singular values is performed
         by using the square-root method of [2].

References

      [1] Safonov, M.G., Jonckheere, E.A., Verma, M. and Limebeer, D.J.
          Synthesis of positive real multivariable feedback systems,
          Int. J. Control, Vol. 45, pp. 817-842, 1987.

      [2] 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.

Numerical Aspects

      The implemented method relies on a square-root technique.
                                      3
      The algorithms require about 17N  floating point operations.

Further Comments

      None

Example

Program Text

    *     AB13AD 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
          PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX )
          INTEGER          LDWORK
          PARAMETER        ( LDWORK = NMAX*( MAX( NMAX, MMAX, PMAX ) + 5 )
         $                        + ( NMAX*( NMAX + 1 ) )/2 )
    *     .. Local Scalars ..
          DOUBLE PRECISION ALPHA, SHNORM
          INTEGER          I, INFO, J, M, N, NS, P
          CHARACTER*1      DICO, EQUIL
    *     .. Local Arrays ..
          DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
         $                 DWORK(LDWORK), HSV(NMAX)
    *     .. External Functions ..
          DOUBLE PRECISION AB13AD
          EXTERNAL         AB13AD
    *     .. 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, DICO, EQUIL
          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 )
    *              Compute the Hankel-norm of the ALPHA-stable projection of
    *              (A,B,C).
                   SHNORM = AB13AD( DICO, EQUIL, N, M, P, ALPHA, A, LDA, B,
         $                          LDB, C, LDC, NS, HSV, DWORK, LDWORK,
         $                          INFO)
    *
                   IF ( INFO.NE.0 ) THEN
                      WRITE ( NOUT, FMT = 99998 ) INFO
                   ELSE
                      WRITE ( NOUT, FMT = 99997 ) SHNORM
                      WRITE ( NOUT, FMT = 99987 )
                      WRITE ( NOUT, FMT = 99995 ) ( HSV(J), J = 1,NS )
                   END IF
                END IF
             END IF
          END IF
          STOP
    *
    99999 FORMAT (' AB13AD EXAMPLE PROGRAM RESULTS',/1X)
    99998 FORMAT (' INFO on exit from AB13AD = ',I2)
    99997 FORMAT (' The Hankel-norm of the ALPHA-projection = ',1PD14.5)
    99995 FORMAT (20(1X,F8.4))
    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 ALPHA-projection are')
          END

Program Data

     AB13AD EXAMPLE PROGRAM DATA (Continuous system)
      7  2  3   0.0  C  N
     -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

Program Results

     AB13AD EXAMPLE PROGRAM RESULTS

     The Hankel-norm of the ALPHA-projection =    2.51388D+00

     The Hankel singular values of ALPHA-projection are
       2.5139   2.0846   1.9178   0.7666   0.5473   0.0253   0.0246

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