AB13HD

L-infinity norm of a state space system in standard or in descriptor form

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

Purpose

  To compute the L-infinity norm of a proper continuous-time or
  causal discrete-time system, either standard or in the descriptor
  form,

                                  -1
     G(lambda) = C*( lambda*E - A ) *B + D .

  The norm is finite if and only if the matrix pair (A,E) has no
  finite eigenvalue on the boundary of the stability domain, i.e.,
  the imaginary axis, or the unit circle, respectively.

Specification
      SUBROUTINE AB13HD( DICO, JOBE, EQUIL, JOBD, CKPROP, REDUCE, POLES,
     $                   N, M, P, RANKE, FPEAK, A, LDA, E, LDE, B, LDB,
     $                   C, LDC, D, LDD, NR, GPEAK, TOL, IWORK, DWORK,
     $                   LDWORK, ZWORK, LZWORK, BWORK, IWARN, INFO )
C     .. Scalar Arguments ..
      CHARACTER          CKPROP, DICO, EQUIL, JOBD, JOBE, POLES, REDUCE
      INTEGER            INFO, IWARN, LDA, LDB, LDC, LDD, LDE, LDWORK,
     $                   LZWORK, M, N, NR, P, RANKE
C     .. Array Arguments ..
      COMPLEX*16         ZWORK(  * )
      DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, * ),
     $                   D( LDD, * ), DWORK(  * ), E( LDE, * ),
     $                   FPEAK(  2 ), GPEAK(  2 ), TOL( * )
      INTEGER            IWORK(  * )
      LOGICAL            BWORK(  * )

Arguments

Mode Parameters

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

  JOBE    CHARACTER*1
          Specifies whether E is an identity matrix, a general
          square matrix, or a matrix in compressed form, as follows:
          = 'I':  E is the identity matrix;
          = 'G':  E is a general matrix;
          = 'C':  E is in compressed form, i.e., E = [ T  0 ],
                                                     [ 0  0 ]
                  with a square full-rank matrix T.

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

  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;
          = 'F':  D is known to be well-conditioned (hence, to have
                  full rank), for DICO = 'C' and JOBE = 'I'.
          The options JOBD = 'D' and JOBD = 'F' produce the same
          results, but much less memory is needed for JOBD = 'F'.

  CKPROP  CHARACTER*1
          If DICO = 'C' and JOBE <> 'I', specifies whether the user
          wishes to check the properness of the transfer function of
          the descriptor system, as follows:
          = 'C':  check the properness;
          = 'N':  do not check the properness.
          If the test is requested and the system is found improper
          then GPEAK and FPEAK are both set to infinity, i.e., their
          second component is zero; in addition, IWARN is set to 2.
          If the test is not requested, but the system is improper,
          the resulted GPEAK and FPEAK may be wrong.
          If DICO = 'D' or JOBE = 'I', this option is ineffective.

  REDUCE  CHARACTER*1
          If CKPROP = 'C', specifies whether the user wishes to
          reduce the system order, by removing all uncontrollable
          and unobservable poles before computing the norm, as
          follows:
          = 'R': reduce the system order;
          = 'N': compute the norm without reducing the order.
          If CKPROP = 'N', this option is ineffective.

  POLES   CHARACTER*1
          Specifies whether the user wishes to use all or part of
          the poles to compute the test frequencies (in the non-
          iterative part of the algorithm), or all or part of the
          midpoints (in the iterative part of the algorithm), as
          follows:
          = 'A': use all poles with non-negative imaginary parts
                 and all midpoints;
          = 'P': use part of the poles and midpoints.

Input/Output Parameters
  N       (input) INTEGER
          The order of the system.  N >= 0.

  M       (input) INTEGER
          The column size of the matrix B.  M >= 0.

  P       (input) INTEGER
          The row size of the matrix C.  P >= 0.

  RANKE   (input) INTEGER
          If JOBE = 'C', RANKE denotes the rank of the descriptor
          matrix E or the size of the full-rank block T.
          0 <= RANKE <= N.

  FPEAK   (input/output) DOUBLE PRECISION array, dimension (2)
          On entry, this parameter must contain an estimate of the
          frequency where the gain of the frequency response would
          achieve its peak value. Setting FPEAK(2) = 0 indicates an
          infinite frequency. An accurate estimate could reduce the
          number of iterations of the iterative algorithm. If no
          estimate is available, set FPEAK(1) = 0, and FPEAK(2) = 1.
          FPEAK(1) >= 0, FPEAK(2) >= 0.
          On exit, if INFO = 0, this array contains the frequency
          OMEGA, where the gain of the frequency response achieves
          its peak value GPEAK, i.e.,

              || G ( j*OMEGA ) || = GPEAK ,  if DICO = 'C', or

                      j*OMEGA
              || G ( e       ) || = GPEAK ,  if DICO = 'D',

          where OMEGA = FPEAK(1), if FPEAK(2) > 0, and OMEGA is
          infinite, if FPEAK(2) = 0. (If nonzero, FPEAK(2) = 1.)
          For discrete-time systems, it is assumed that the sampling
          period is Ts = 1. If Ts <> 1, the frequency corresponding
          to the peak gain is OMEGA/Ts.

  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 EQUIL = 'S' and CKPROP = 'N', the leading
          N-by-N part of this array contains the state dynamics
          matrix of an equivalent, scaled system.
          On exit, if CKPROP = 'C', DICO = 'C', and JOBE <> 'I', the
          leading NR-by-NR part of this array contains the state
          dynamics matrix of an equivalent reduced, possibly scaled
          (if EQUIL = 'S') system, used to check the properness.
          Otherwise, the array A is unchanged.

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

  E       (input/output) DOUBLE PRECISION array, dimension (LDE,K),
          where K is N, RANKE, or 0, if JOBE = 'G', 'C', or 'I',
          respectively.
          On entry, if JOBE = 'G', the leading N-by-N part of this
          array must contain the descriptor matrix E of the system.
          If JOBE = 'C', the leading RANKE-by-RANKE part of this
          array must contain the full-rank block T of the descriptor
          matrix E.
          If JOBE = 'I', then E is assumed to be the identity matrix
          and is not referenced.
          On exit, if EQUIL = 'S' and CKPROP = 'N', the leading
          K-by-K part of this array contains the descriptor matrix
          of an equivalent, scaled system.
          On exit, if CKPROP = 'C', DICO = 'C', and JOBE <> 'I', the
          leading MIN(K,NR)-by-MIN(K,NR) part of this array contains
          the descriptor matrix of an equivalent reduced, possibly
          scaled (if EQUIL = 'S') system, used to check the
          properness.
          Otherwise, the array E is unchanged.

  LDE     INTEGER
          The leading dimension of the array E.
          LDE >= MAX(1,N),     if JOBE = 'G';
          LDE >= MAX(1,RANKE), if JOBE = 'C';
          LDE >= 1,            if JOBE = 'I'.

  B       (input/output) DOUBLE PRECISION array, dimension (LDB,M)
          On entry, the leading N-by-M part of this array must
          contain the system input matrix B.
          On exit, if EQUIL = 'S' and CKPROP = 'N', the leading
          NR-by-M part of this array contains the system input
          matrix of an equivalent, scaled system.
          On exit, if CKPROP = 'C', DICO = 'C', and JOBE <> 'I', the
          leading NR-by-M part of this array contains the system
          input matrix of an equivalent reduced, possibly scaled (if
          EQUIL = 'S') system, used to check the properness.
          Otherwise, the array B is unchanged.

  LDB     INTEGER
          The leading dimension of the 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 system output matrix C.
          On exit, if EQUIL = 'S' and CKPROP = 'N', the leading
          P-by-NR part of this array contains the system output
          matrix of an equivalent, scaled system.
          On exit, if CKPROP = 'C', DICO = 'C', and JOBE <> 'I', the
          leading P-by-NR part of this array contains the system
          output matrix of an equivalent reduced, possibly scaled
          (if EQUIL = 'S') system, used to check the properness.
          Otherwise, the array C is unchanged.

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

  D       (input) DOUBLE PRECISION array, dimension (LDD,M)
          If JOBD = 'D' or JOBD = 'F', the leading P-by-M part of
          this array must contain the direct 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' or JOBD = 'F';
          LDD >= 1,        if JOBD = 'Z'.

  NR      (output) INTEGER
          If CKPROP = 'C', DICO = 'C', and JOBE <> 'I', the order of
          the reduced system. Otherwise, NR = N.

  GPEAK   (output) DOUBLE PRECISION array, dimension (2)
          The L-infinity norm of the system, i.e., the peak gain
          of the frequency response (as measured by the largest
          singular value in the MIMO case), coded in the same way
          as FPEAK.

Tolerances
  TOL     DOUBLE PRECISION array, dimension K, where K = 2, if
          CKPROP = 'N' or DICO = 'D' or JOBE = 'I', and K = 4,
          otherwise.
          TOL(1) is the tolerance used to set the accuracy in
          determining the norm.  0 <= TOL(1) < 1.
          TOL(2) is the threshold value for magnitude of the matrix
          elements, if EQUIL = 'S': elements with magnitude less
          than or equal to TOL(2) are ignored for scaling. If the
          user sets TOL(2) >= 0, then the given value of TOL(2) is
          used. If the user sets TOL(2) < 0, then an implicitly
          computed, default threshold, THRESH, is used instead,
          defined by THRESH = 0.1, if MN/MX < EPS, and otherwise,
          THRESH = MIN( 100*(MN/(EPS**0.25*MX))**0.5, 0.1 ), where
          MX and MN are the maximum and the minimum nonzero absolute
          value, respectively, of the elements of A and E, and EPS
          is the machine precision (see LAPACK Library routine
          DLAMCH). TOL(2) = 0 is not always a good choice.
          TOL(2) < 1. TOL(2) is not used if EQUIL = 'N'.
          TOL(3) is the tolerance to be used in rank determinations
          when transforming (lambda*E-A,B,C), if CKPROP = 'C'. If
          the user sets TOL(3) > 0, then the given value of TOL(3)
          is used as a lower bound for reciprocal condition numbers
          in rank determinations; a (sub)matrix whose estimated
          condition number is less than 1/TOL(3) is considered to be
          of full rank.  If the user sets TOL(3) <= 0, then an
          implicitly computed, default tolerance, defined by
          TOLDEF1 = N*N*EPS, is used instead.  TOL(3) < 1.
          TOL(4) is the tolerance to be used for checking the
          singularity of the matrices A and E when CKPROP = 'C'.
          If the user sets TOL(4) > 0, then the given value of
          TOL(4) is used.  If the user sets TOL(4) <= 0, then an
          implicitly computed, default tolerance, defined by
          TOLDEF2 = N*EPS, is used instead. The 1-norms of A and E
          are also taken into account.  TOL(4) < 1.

Workspace
  IWORK   INTEGER array, dimension (LIWORK)
          LIWORK >= 1, if MIN(N,P,M) = 0, or B = 0, or C = 0; else
          LIWORK >= MAX(1,N), if DICO = 'C', JOBE = 'I', and
                    JOBD <> 'D';
          LIWORK >= 2*N + M + P + R + 12, otherwise, where
                    R = 0, if M + P is even,
                    R = 1, if M + P is odd.
          On exit, if INFO = 0, IWORK(1) returns the number of
          iterations performed by the iterative algorithm
          (possibly 0).

  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
          On exit, if INFO = 0, DWORK(1) contains the optimal value
          of LDWORK.
          On exit, if  INFO = -28,  DWORK(1)  returns the minimum
          value of LDWORK. These values are also set when LDWORK = 0
          on entry, but no error message related to LDWORK is issued
          by XERBLA.

  LDWORK  INTEGER
          The dimension of the array DWORK.
          LDWORK >= 1, if MIN(M,P) = 0 or ( JOBD = 'Z' and
                                      ( N = 0 or B = 0 or C = 0 ) );
          LDWORK >= P*M + x, if ( ( N = 0 and MIN(M,P) > 0 )
                       or ( B = 0 or C = 0 ) ) and JOBD <> 'Z',
                    where
                    x = MAX( 4*MIN(M,P) + MAX(M,P), 6*MIN(M,P) ),
                                    if DICO = 'C',
                    x = 6*MIN(M,P), if DICO = 'D';
          LDWORK >= MAX( 1, N*(N+M+P+2) + MAX( N*(N+M+2) + P*M + x,
                                               4*N*N + 9*N ) ),
                    if DICO = 'C', JOBE = 'I' and JOBD = 'Z'.
          LDWORK >= MAX( 1, (N+M)*(M+P) + P*P + x,
                         2*N*(N+M+P+1) + N + MIN(P,M) +
                         MAX( M*(N+P) + N + x, N*N +
                              MAX( N*(P+M) + MAX(M,P),
                                   2*N*N + 8*N ) ) ),
                    if DICO = 'C', JOBE = 'I' and JOBD = 'F'.
          The formulas for other cases, e.g., for JOBE <> 'I' or
          CKPROP = 'C', contain additional and/or other terms.
          The minimum value of LDWORK for all cases can be obtained
          in DWORK(1) when LDWORK is set to 0 on entry.
          For good performance, LDWORK must generally be larger.

          If LDWORK = -1, then a workspace query is assumed;
          the routine only calculates the optimal size of the
          DWORK array, returns this value as the first entry of
          the DWORK array, and no error message related to LDWORK
          is issued by XERBLA.

  ZWORK   COMPLEX*16 array, dimension (LZWORK)
          On exit, if INFO = 0, ZWORK(1) contains the optimal
          LZWORK.
          On exit, if  INFO = -30,  ZWORK(1)  returns the minimum
          value of LZWORK. These values are also set when LZWORK = 0
          on entry, but no error message related to LZWORK is issued
          by XERBLA.
          If LDWORK = 0 and LZWORK = 0 are both set on entry, then
          on exit, INFO = -30, but both DWORK(1) and ZWORK(1) are
          set the minimum values of LDWORK and LZWORK, respectively.

  LZWORK  INTEGER
          The dimension of the array ZWORK.
          LZWORK >= 1,  if MIN(N,M,P) = 0, or B = 0, or C = 0;
          LZWORK >= MAX(1, (N+M)*(N+P) + 2*MIN(M,P) + MAX(M,P)),
                        otherwise.
          For good performance, LZWORK must generally be larger.

          If LZWORK = -1, then a workspace query is assumed;
          the routine only calculates the optimal size of the
          ZWORK array, returns this value as the first entry of
          the ZWORK array, and no error message related to LZWORK
          is issued by XERBLA.

  BWORK   LOGICAL array, dimension (N)

Warning Indicator
  IWARN   INTEGER
          = 0:  no warning;
          = 1:  the descriptor system is singular. GPEAK(1) and
                GPEAK(2) are set to 0. FPEAK(1) and FPEAK(2) are
                set to 0 and 1, respectively;
          = 2:  the descriptor system is improper. GPEAK(1) and
                GPEAK(2) are set to 1 and 0, respectively,
                corresponding to infinity. FPEAK(1) and FPEAK(2) are
                set similarly. This warning can only appear if
                CKPROP = 'C'.

Error Indicator
  INFO    INTEGER
          = 0:  successful exit;
          < 0:  if INFO = -i, the i-th argument had an illegal
                value;
          = 1:  a matrix is (numerically) singular or the Sylvester
                equation is very ill-conditioned, when computing the
                largest singular value of G(infinity) (for
                DICO = 'C'); the descriptor system is nearly
                singular; the L-infinity norm could be infinite;
          = 2:  the (periodic) QR (or QZ) algorithm for computing
                eigenvalues did not converge;
          = 3:  the SVD algorithm for computing singular values did
                not converge;
          = 4:  the tolerance is too small and the algorithm did
                not converge; this is a warning; 
          = 5:  other computations than QZ iteration, or reordering
                of eigenvalues, failed in the LAPACK Library
                routines DHGEQZ or DTGSEN, respectively;
          = 6:  the numbers of "finite" eigenvalues before and after
                reordering differ; the threshold used might be
                unsuitable.

Method
  The routine implements the method presented in [2], which is an
  extension of the method in [1] for descriptor systems. There are
  several improvements and refinements [3-5] to increase numerical
  robustness, accuracy and efficiency, such as the usage of
  structure-preserving eigenvalue computations for skew-Hamiltonian/
  Hamiltonian eigenvalue problems in the iterative method in [2].

References
  [1] Bruinsma, N.A. and Steinbuch, M.
      A fast algorithm to compute the H-infinity-norm of a transfer
      function matrix.
      Systems & Control Letters, vol. 14, pp. 287-293, 1990.

  [2] Voigt, M.
      L-infinity-Norm Computation for Descriptor Systems.
      Diploma Thesis, Fakultaet fuer Mathematik, TU Chemnitz,
      http://nbn-resolving.de/urn:nbn:de:bsz:ch1-201001050.

  [3] Benner, P., Sima, V. and Voigt, M.
      L-infinity-norm computation for continuous-time descriptor
      systems using structured matrix pencils.
      IEEE Trans. Auto. Contr., AC-57, pp.233-238, 2012.

  [4] Benner, P., Sima, V. and Voigt, M.
      Robust and efficient algorithms for L-infinity-norm
      computations for descriptor systems.
      7th IFAC Symposium on Robust Control Design (ROCOND'12),
      pp. 189-194, 2012.

  [5] Benner, P., Sima, V. and Voigt, M.
      Algorithm 961: Fortran 77 subroutines for the solution of
      skew-Hamiltonian/Hamiltonian eigenproblems.
      ACM Trans. Math. Softw, 42, pp. 1-26, 2016.

Numerical Aspects
  If the algorithm does not converge in MAXIT = 30 iterations
  (INFO = 4), the tolerance must be increased, or the system is
  improper.

Further Comments
  Setting POLES = 'P' usually saves some computational effort. The
  number of poles used is defined by the parameters BM, BNEICD,
  BNEICM, BNEICX, BNEIR and SWNEIC.
  Both real and complex optimal workspace sizes are computed if
  either LDWORK = -1 or LZWORK = -1.

Example

Program Text

  None
Program Data
  None
Program Results
  None

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