Purpose
Given the descriptor system (A-lambda*E,B,C) with the system
matrices A, E and B of the form
( A1 X1 ) ( E1 Y1 ) ( B1 B2 )
A = ( ) , E = ( ) , B = ( ) ,
( 0 X2 ) ( 0 Y2 ) ( 0 0 )
where
- B is an L-by-(M1+M2) matrix,
with B1 an N1-by-M1 submatrix, B2 an N1-by-M2 submatrix,
- A is an L-by-N matrix, with A1 an N1-by-N1 submatrix,
- E is an L-by-N matrix, with E1 an N1-by-N1 submatrix
with LBE nonzero sub-diagonals,
this routine reduces the pair (A1-lambda*E1,[B1 B2]) to the form
Qc'*[B1 B2 A1-lambda*E1 ]*diag(I,Zc) =
( Bc1 Bc2 Ac-lambda*Ec * )
( ) ,
( 0 0 0 Anc-lambda*Enc )
where:
1) the pencil ( Bc1 Bc2 Ac-lambda*Ec ) has full row rank NR for
all finite lambda and is in a staircase form with
[ A11 A12 . . . A1,p-2 A1,p-1 A1p ]
[ A21 A22 . . . A2,p-2 A2,p-1 A2p ]
[ A31 A32 . . . A3,p-2 A3,p-1 A3p ]
[ 0 A42 . . . A4,p-2 A4,p-1 A4p ]
Ac = [ . . . . . . . . ], (1)
[ . . . . . . . ]
[ . . . . . . ]
[ 0 0 . . . Ap,p-2 Ap,p-1 App ]
[ A1,-1 A1,0 ]
[ 0 A2,0 ]
[ 0 0 ] ( E11 E12 ... E1p )
[ 0 0 ] ( 0 E22 ... E2p )
[Bc1 Bc2] = [ . . ], Ec = ( . . . . ),
[ . . ] ( . . . . )
[ . . ] ( 0 0 ... Epp )
[ 0 0 ]
where the block Ai,i-2 is an rtau(i)-by-rtau(i-2) full row
rank matrix (with rtau(-1) = M1, rtau(0) = M2) and Ei,i is an
rtau(i)-by-rtau(i) upper triangular matrix.
2) the pencil Anc-lambda*Enc is regular of order N1-NR with Enc
upper triangular; this pencil contains the uncontrollable
finite eigenvalues of the pencil (A1-lambda*E1).
The transformations are applied to the whole matrices A, E, B
and C. The left and/or right orthogonal transformations Qc and Zc,
performed to reduce the pencil, can be optionally accumulated in
the matrices Q and Z, respectively.
The reduced order descriptor system (Ac-lambda*Ec,Bc,Cc) has no
uncontrollable finite eigenvalues and has the same transfer-
function matrix as the original system (A-lambda*E,B,C).
Specification
SUBROUTINE TG01HU( COMPQ, COMPZ, L, N, M1, M2, P, N1, LBE, A, LDA,
$ E, LDE, B, LDB, C, LDC, Q, LDQ, Z, LDZ, NR,
$ NRBLCK, RTAU, TOL, IWORK, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER COMPQ, COMPZ
INTEGER INFO, L, LBE, LDA, LDB, LDC, LDE, LDQ, LDWORK,
$ LDZ, M1, M2, N, N1, NR, NRBLCK, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER IWORK( * ), RTAU( * )
DOUBLE PRECISION A( LDA, * ), B( LDB, * ), C( LDC, * ),
$ DWORK( * ), E( LDE, * ), Q( LDQ, * ),
$ Z( LDZ, * )
Arguments
Mode Parameters
COMPQ CHARACTER*1
= 'N': do not compute Q;
= 'I': Q is initialized to the unit matrix, and the
orthogonal matrix Q is returned;
= 'U': Q must contain an orthogonal matrix Q1 on entry,
and the product Q1*Q is returned.
COMPZ CHARACTER*1
= 'N': do not compute Z;
= 'I': Z is initialized to the unit matrix, and the
orthogonal matrix Z is returned;
= 'U': Z must contain an orthogonal matrix Z1 on entry,
and the product Z1*Z is returned.
Input/Output Parameters
L (input) INTEGER
The number of descriptor state equations; also the number
of rows of the matrices A, E and B. L >= 0.
N (input) INTEGER
The dimension of the descriptor state vector; also the
number of columns of the matrices A, E and C. N >= 0.
M1 (input) INTEGER
The number of system inputs in U1, or of columns of B1.
M1 >= 0.
M2 (input) INTEGER
The number of system inputs in U2, or of columns of B2.
M2 >= 0.
P (input) INTEGER
The dimension of descriptor system output; also the
number of rows of the matrix C. P >= 0.
N1 (input) INTEGER
The order of the subsystem (A1-lambda*E1,B1,C1) to be
reduced. MIN(L,N) >= N1 >= 0.
LBE (input) INTEGER
The number of nonzero sub-diagonals of the submatrix E1.
MAX(0,N1-1) >= LBE >= 0.
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading L-by-N part of this array must
contain the L-by-N state matrix A in the partitioned form
( A1 X1 )
A = ( ) ,
( 0 X2 )
where A1 is N1-by-N1.
On exit, the leading L-by-N part of this array contains
the transformed state matrix,
( Ac * * )
Qc'*A*diag(Zc,I) = ( 0 Anc * ) ,
( 0 0 * )
where Ac is NR-by-NR and Anc is (N1-NR)-by-(N1-NR).
The matrix ( Bc Ac ) is in the controllability staircase
form (1).
LDA INTEGER
The leading dimension of the array A. LDA >= MAX(1,L).
E (input/output) DOUBLE PRECISION array, dimension (LDE,N)
On entry, the leading L-by-N part of this array must
contain the L-by-N descriptor matrix E in the partitioned
form
( E1 Y1 )
E = ( ) ,
( 0 Y2 )
where E1 is an N1-by-N1 matrix with LBE nonzero
sub-diagonals.
On exit, the leading L-by-N part of this array contains
the transformed descriptor matrix
( Ec * * )
Qc'*E*diag(Zc,I) = ( 0 Enc * ) ,
( 0 0 * )
where Ec is NR-by-NR and Enc is (N1-NR)-by-(N1-NR).
Both Ec and Enc are upper triangular and Enc is
nonsingular.
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,L).
B (input/output) DOUBLE PRECISION array, dimension (LDB,M)
with M = M1 + M2.
On entry, the leading L-by-M part of this array must
contain the L-by-M input matrix B in the partitioned form
( Bi )
B = ( ) ,
( 0 )
where Bi is N1-by-M.
On exit, the leading L-by-M part of this array contains
the transformed input matrix
( Bc )
Qc'*B = ( ) ,
( 0 )
where Bc is NR-by-M.
The matrix ( Bc Ac ) is in the controllability staircase
form (1).
LDB INTEGER
The leading dimension of the array B. LDB >= MAX(1,L).
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the state/output matrix C.
On exit, the leading P-by-N part of this array contains
the transformed matrix C*Zc.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,P).
Q (input/output) DOUBLE PRECISION array, dimension (LDQ,L)
If COMPQ = 'N': Q is not referenced.
If COMPQ = 'I': on entry, Q need not be set;
on exit, the leading L-by-L part of this
array contains the orthogonal matrix Qc,
where Qc' is the product of the
transformations applied to A, E, and B on
the left.
If COMPQ = 'U': on entry, the leading L-by-L part of this
array must contain an orthogonal matrix Q;
on exit, the leading L-by-L part of this
array contains the orthogonal matrix
Q*Qc.
LDQ INTEGER
The leading dimension of the array Q.
LDQ >= 1, if COMPQ = 'N';
LDQ >= MAX(1,L), if COMPQ = 'I' or 'U'.
Z (input/output) DOUBLE PRECISION array, dimension (LDZ,N)
If COMPZ = 'N': Z is not referenced.
If COMPZ = 'I': on entry, Z need not be set;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix Zc,
i.e., the product of the transformations
applied to A, E, and C on the right.
If COMPZ = 'U': on entry, the leading N-by-N part of this
array must contain an orthogonal matrix Z;
on exit, the leading N-by-N part of this
array contains the orthogonal matrix
Z*Zc.
LDZ INTEGER
The leading dimension of the array Z.
LDZ >= 1, if COMPZ = 'N';
LDZ >= MAX(1,N), if COMPZ = 'I' or 'U'.
NR (output) INTEGER
The order of the reduced matrices Ac and Ec, and the
number of rows of the reduced matrix Bc; also the order of
the controllable part of the pair (B, A-lambda*E).
NRBLCK (output) INTEGER
The number p, of full row rank blocks Ai,i-2 in the
staircase form of the pencil (Bc1 Bc2 Ac-lambda*Ec).
RTAU (output) INTEGER array, dimension (2*N1)
The leading NRBLCK elements of this array contain the
orders of the diagonal blocks of Ac. NRBLCK is always
an even number, and the NRBLCK/2 odd and even components
of RTAU have decreasing values, respectively.
Note that some elements of RTAU can be zero.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in rank determinations when
transforming (A-lambda*E, B). If the user sets TOL > 0,
then the given value of TOL 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 is considered to be of full rank. If the user sets
TOL <= 0, then an implicitly computed, default tolerance,
defined by TOLDEF = L*N*EPS, is used instead, where
EPS is the machine precision (see LAPACK Library routine
DLAMCH). TOL < 1.
Workspace
IWORK INTEGER array, dimension (M)
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 >= 1, if MIN(N1,M) = 0; otherwise,
LDWORK >= MAX(N1+MAX(L,N,M),2*M), if LBE > 0 and N1 > 2;
LDWORK >= MAX(1,L,N,2*M), if LBE = 0 or N1 <= 2.
For optimum performance LDWORK should 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.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
Method
The implemented algorithm [1] represents a specialization of the controllability staircase algorithm of [2] to the special structure of the input matrix B = [B1,B2].References
[1] Varga, A.
Reliable algorithms for computing minimal dynamic covers for
descriptor systems.
Proc. of MTNS'04, Leuven, Belgium, 2004.
[2] Varga, A.
Computation of Irreducible Generalized State-Space
Realizations.
Kybernetika, vol. 26, pp. 89-106, 1990.
Numerical Aspects
The algorithm is numerically backward stable and requires 0( N*N1**2 ) floating point operations.Further Comments
NoneExample
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