## 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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