Purpose
To extract from the system pencil
( A-lambda*I B )
S(lambda) = ( )
( C D )
a regular pencil Af-lambda*Ef which has the finite Smith zeros of
S(lambda) as generalized eigenvalues. The routine also computes
the orders of the infinite Smith zeros and determines the singular
and infinite Kronecker structure of the system pencil, i.e., the
right and left Kronecker indices, and the multiplicities of the
infinite eigenvalues.
Specification
SUBROUTINE AB08NW( EQUIL, N, M, P, A, LDA, B, LDB, C, LDC, D, LDD,
$ NFZ, NRANK, NIZ, DINFZ, NKROR, NINFE, NKROL,
$ INFZ, KRONR, INFE, KRONL, E, LDE, TOL, IWORK,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER EQUIL
INTEGER DINFZ, INFO, LDA, LDB, LDC, LDD, LDE, LDWORK,
$ M, N, NFZ, NINFE, NIZ, NKROL, NKROR, NRANK, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
INTEGER INFE(*), INFZ(*), IWORK(*), KRONL(*), KRONR(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DWORK(*), E(LDE,*)
Arguments
Mode Parameters
EQUIL CHARACTER*1
Specifies whether the user wishes to balance the system
matrix as follows:
= 'S': Perform balancing (scaling);
= 'N': Do not perform balancing.
Input/Output Parameters
N (input) INTEGER.
The order of the square matrix A, the number of rows of
the matrix B, and number of columns of the matrix C.
N >= 0.
M (input) INTEGER.
The number of columns of the matrix B. M >= 0.
P (input) INTEGER.
The number of rows of the matrix C. P >= 0.
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 of the system.
On exit, the leading NFZ-by-NFZ part of this array
contains the matrix Af of the reduced pencil.
LDA INTEGER
The leading dimension of the 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 input/state matrix B of the system.
On exit, this matrix does not contain useful information.
LDB INTEGER
The leading dimension of the array B. LDB >= 1, and
LDB >= MAX(1,N), if M > 0.
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 of the system.
On exit, this matrix does not contain useful information.
LDC INTEGER
The leading dimension of the array C. LDC >= MAX(1,P).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading P-by-M part of this array must contain the
direct transmission matrix D of the system.
LDD INTEGER
The leading dimension of the array D. LDD >= MAX(1,P).
NFZ (output) INTEGER
The number of finite zeros.
NRANK (output) INTEGER
The normal rank of the system pencil.
NIZ (output) INTEGER
The number of infinite zeros.
DINFZ (output) INTEGER
The maximal multiplicity of infinite Smith zeros.
NKROR (output) INTEGER
The number of right Kronecker indices.
NINFE (output) INTEGER
The number of elementary infinite blocks.
NKROL (output) INTEGER
The number of left Kronecker indices.
INFZ (output) INTEGER array, dimension (N+1)
The leading DINFZ elements of INFZ contain information
on the infinite elementary divisors as follows:
the system has INFZ(i) infinite elementary divisors in
the Smith form of degree i, where i = 1,2,...,DINFZ.
KRONR (output) INTEGER array, dimension (N+1)
The leading NKROR elements of this array contain the
right Kronecker (column) indices.
INFE (output) INTEGER array, dimension (N+1)
The leading NINFE elements of INFE contain the
multiplicities of infinite eigenvalues.
KRONL (output) INTEGER array, dimension (N+1)
The leading NKROL elements of this array contain the
left Kronecker (row) indices.
E (output) DOUBLE PRECISION array, dimension (LDE,N)
The leading NFZ-by-NFZ part of this array contains the
matrix Ef of the reduced pencil.
LDE INTEGER
The leading dimension of the array E. LDE >= MAX(1,N).
Tolerances
TOL DOUBLE PRECISION
A tolerance used in rank decisions to determine the
effective rank, which is defined as the order of the
largest leading (or trailing) triangular submatrix in the
QR (or RQ) factorization with column (or row) pivoting
whose estimated condition number is less than 1/TOL.
If the user sets TOL <= 0, then an implicitly computed,
default tolerance TOLDEF = MAX(N+P,N+M)**2*EPS, is used
instead, where EPS is the machine precision (see LAPACK
Library routine DLAMCH). TOL < 1.
Workspace
IWORK INTEGER array, dimension (MAX(M,P))
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 MAX(N,M,P) = 0; otherwise,
LDWORK >= MAX( MIN(P,M) + M + MAX(2*M,N) - 1,
MIN(P,N) + MAX(N + MAX(P,M), 3*P - 1 ) ) +
MAX(P+N,M+N)*MAX(P+N,M+N).
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 routine extracts from the system matrix of a state space
system, (A-lambda*I,B,C,D), a regular pencil Af-lambda*Ef, which
has the finite zeros of the system as generalized eigenvalues.
The procedure has the following main computational steps:
(a) construct the (N+P)-by-(M+N) system pencil
S(lambda) = (B A)-lambda*( 0 I );
(D C) ( 0 0 )
(b) reduce S(lambda) to S1(lambda) with the same finite zeros
and right Kronecker structure, but with D of full row rank;
(c) reduce the pencil S1(lambda) to S2(lambda) with the same
finite zeros and with D square invertible;
(d) perform a unitary transformation on the columns of
S2(lambda) = (A-lambda*I B), in order to reduce it to
( C D)
(Af-lambda*Ef X), with Y and Ef square invertible;
( 0 Y)
(e) compute the right and left Kronecker indices of the system
matrix, which, together with the multiplicities of the
finite and infinite eigenvalues, constitute the complete
set of structural invariants under strict equivalence
transformations of a linear system.
References
[1] Svaricek, F.
Computation of the Structural Invariants of Linear
Multivariable Systems with an Extended Version of the
Program ZEROS.
System & Control Letters, 6, pp. 261-266, 1985.
[2] Emami-Naeini, A. and Van Dooren, P.
Computation of Zeros of Linear Multivariable Systems.
Automatica, 18, pp. 415-430, 1982.
Numerical Aspects
The algorithm is backward stable (see [2] and [1]).Further Comments
In order to compute the finite Smith zeros of the system explicitly, a call to this routine may be followed by a call to the LAPACK Library routine DGGEV.Example
Program Text
* AB08NW EXAMPLE PROGRAM TEXT
* Copyright (c) 2002-2017 NICONET e.V.
*
* .. Parameters ..
DOUBLE PRECISION ZERO
PARAMETER ( ZERO = 0.0D0 )
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, MMAX, PMAX
PARAMETER ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER MPMAX
PARAMETER ( MPMAX = MAX( MMAX, PMAX ) )
INTEGER LDA, LDB, LDC, LDD, LDE, LDQ, LDZ
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDD = PMAX, LDE = NMAX, LDQ = 1, LDZ = 1 )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( MAX( MIN( PMAX, MMAX ) + MMAX +
$ MAX( 2*MMAX, NMAX ) - 1,
$ MIN( PMAX, NMAX ) +
$ MAX( NMAX + MPMAX,
$ 3*PMAX - 1 ) ) +
$ ( NMAX + MPMAX )**2, 8*NMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER DINFZ, I, INFO, J, M, N, NFZ, NINFE, NIZ, NKROL,
$ NKROR, NRANK, P
CHARACTER*1 EQUIL
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), ALFI(NMAX), ALFR(NMAX),
$ AS(LDA,NMAX), B(LDB,MMAX), BS(LDB,MMAX),
$ BETA(NMAX), C(LDC,NMAX), CS(LDC,NMAX),
$ D(LDD,MMAX), DWORK(LDWORK), E(LDE,NMAX),
$ Q(LDQ,1), Z(LDZ,1)
INTEGER INFE(NMAX+1), INFZ(NMAX+1), IWORK(MPMAX),
$ KRONL(NMAX+1), KRONR(NMAX+1)
* .. External Subroutines ..
EXTERNAL AB08NW, DLACPY, DGEGV
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN
* .. 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, TOL, EQUIL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99972 ) 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 = 99971 ) 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 = 99970 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
* Save the matrices A, B, and C.
CALL DLACPY( 'Full', N, N, A, LDA, AS, LDA )
CALL DLACPY( 'Full', N, M, B, LDB, BS, LDB )
CALL DLACPY( 'Full', P, N, C, LDC, CS, LDC )
* Check the observability and compute the ordered set of
* the observability indices (call the routine with M = 0).
CALL AB08NW( EQUIL, N, 0, P, A, LDA, B, LDB, C, LDC, D,
$ LDD, NFZ, NRANK, NIZ, DINFZ, NKROR, NINFE,
$ NKROL, INFZ, KRONR, INFE, KRONL, E, LDE,
$ TOL, IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99994 ) ( KRONL(I), I = 1,P )
IF ( NFZ.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99993 )
ELSE
WRITE ( NOUT, FMT = 99992 ) N - NFZ
WRITE ( NOUT, FMT = 99991 )
WRITE ( NOUT, FMT = 99990 )
DO 20 I = 1, NFZ
WRITE ( NOUT, FMT = 99989 )
$ ( A(I,J), J = 1,NFZ )
20 CONTINUE
END IF
END IF
* Restore the matrices A and C.
CALL DLACPY( 'Full', N, N, AS, LDA, A, LDA )
CALL DLACPY( 'Full', P, N, CS, LDC, C, LDC )
* Check the controllability and compute the ordered set of
* the controllability indices (call the routine with P = 0)
CALL AB08NW( EQUIL, N, M, 0, A, LDA, B, LDB, C, LDC, D,
$ LDD, NFZ, NRANK, NIZ, DINFZ, NKROR, NINFE,
$ NKROL, INFZ, KRONR, INFE, KRONL, E, LDE,
$ TOL, IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99988 ) ( KRONR(I), I = 1,M )
IF ( NFZ.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99987 )
ELSE
WRITE ( NOUT, FMT = 99986 ) N - NFZ
WRITE ( NOUT, FMT = 99985 )
WRITE ( NOUT, FMT = 99990 )
DO 40 I = 1, NFZ
WRITE ( NOUT, FMT = 99989 )
$ ( A(I,J), J = 1,NFZ )
40 CONTINUE
END IF
END IF
* Restore the matrices A and B.
CALL DLACPY( 'Full', N, N, AS, LDA, A, LDA )
CALL DLACPY( 'Full', N, M, BS, LDB, B, LDB )
* Compute the structural invariants of the given system.
CALL AB08NW( EQUIL, N, M, P, A, LDA, B, LDB, C, LDC, D,
$ LDD, NFZ, NRANK, NIZ, DINFZ, NKROR, NINFE,
$ NKROL, INFZ, KRONR, INFE, KRONL, E, LDE,
$ TOL, IWORK, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99984 ) NFZ
IF ( NFZ.GT.0 ) THEN
* Compute the invariant zeros of the given system.
* Workspace: need 8*NFZ.
WRITE ( NOUT, FMT = 99983 )
CALL DGEGV( 'No vectors', 'No vectors', NFZ, A,
$ LDA, E, LDE, ALFR, ALFI, BETA, Q,
$ LDQ, Z, LDZ, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99997 ) INFO
ELSE
WRITE ( NOUT, FMT = 99981 )
DO 60 I = 1, NFZ
IF ( ALFI(I).EQ.ZERO ) THEN
WRITE ( NOUT, FMT = 99980 )
$ ALFR(I)/BETA(I)
ELSE
WRITE ( NOUT, FMT = 99979 )
$ ALFR(I)/BETA(I),
$ ALFI(I)/BETA(I)
END IF
60 CONTINUE
WRITE ( NOUT, FMT = 99982 )
END IF
END IF
WRITE ( NOUT, FMT = 99978 ) NIZ
IF ( NIZ.GT.0 ) THEN
DO 100 I = 1, DINFZ
WRITE ( NOUT, FMT = 99977 ) INFZ(I), I
100 CONTINUE
END IF
WRITE ( NOUT, FMT = 99976 ) NKROR
IF ( NKROR.GT.0 ) WRITE ( NOUT, FMT = 99975 )
$ ( KRONR(I), I = 1,NKROR )
WRITE ( NOUT, FMT = 99974 ) NKROL
IF ( NKROL.GT.0 ) WRITE ( NOUT, FMT = 99973 )
$ ( KRONL(I), I = 1,NKROL )
IF ( NINFE.GT.0 ) WRITE ( NOUT, FMT = 99969 )
$ ( INFE(I), I = 1,NINFE )
END IF
END IF
END IF
END IF
*
STOP
*
99999 FORMAT (' AB08NW EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB08NW = ',I2)
99997 FORMAT (' INFO on exit from DGEGV = ',I2)
99994 FORMAT (' The left Kronecker indices of (A,C) are ',/(20(I3,2X)))
99993 FORMAT (/' The system (A,C) is completely observable ')
99992 FORMAT (/' The dimension of the observable subspace = ',I3)
99991 FORMAT (/' The output decoupling zeros are the eigenvalues of th',
$ 'e matrix AF. ')
99990 FORMAT (/' The matrix AF is ')
99989 FORMAT (20(1X,F8.4))
99988 FORMAT (//' The right Kronecker indices of (A,B) are ',/(20(I3,2X)
$ ))
99987 FORMAT (/' The system (A,B) is completely controllable ')
99986 FORMAT (/' The dimension of the controllable subspace = ',I3)
99985 FORMAT (/' The input decoupling zeros are the eigenvalues of the',
$ ' matrix AF. ')
99984 FORMAT (//' The number of finite invariant zeros = ',I3)
99983 FORMAT (/' The finite invariant zeros are ')
99982 FORMAT (/' which correspond to the generalized eigenvalues of (l',
$ 'ambda*EF - AF).')
99981 FORMAT (/' real part imag part ')
99980 FORMAT (1X,F9.4)
99979 FORMAT (1X,F9.4,6X,F9.4)
99978 FORMAT (//' The number of infinite zeros = ',I3)
99977 FORMAT ( I4,' infinite zero(s) of order ',I3)
99976 FORMAT (/' The number of right Kronecker indices = ',I3)
99975 FORMAT (/' Right Kronecker (column) indices of (A,B,C,D) are ',
$ /(20(I3,2X)))
99974 FORMAT (/' The number of left Kronecker indices = ',I3)
99973 FORMAT (/' The left Kronecker (row) indices of (A,B,C,D) are ',
$ /(20(I3,2X)))
99972 FORMAT (/' N is out of range.',/' N = ',I5)
99971 FORMAT (/' M is out of range.',/' M = ',I5)
99970 FORMAT (/' P is out of range.',/' P = ',I5)
99969 FORMAT (/' Multiplicities of infinite eigenvalues',/(20(I3,2X)))
END
Program Data
AB08NW EXAMPLE PROGRAM DATA 6 2 3 0.0 N 1.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0 3.0 0.0 0.0 0.0 0.0 0.0 0.0 -4.0 0.0 0.0 0.0 0.0 0.0 0.0 -1.0 0.0 0.0 0.0 0.0 0.0 0.0 3.0 0.0 -1.0 -1.0 0.0 1.0 -1.0 0.0 0.0 0.0 1.0 -1.0 -1.0 1.0 0.0 0.0 1.0 0.0 0.0 0.0 1.0 0.0 1.0 0.0 1.0 0.0 0.0 1.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 0.0 0.0Program Results
AB08NW EXAMPLE PROGRAM RESULTS
The left Kronecker indices of (A,C) are
1 2 2
The dimension of the observable subspace = 5
The output decoupling zeros are the eigenvalues of the matrix AF.
The matrix AF is
-1.0000
The right Kronecker indices of (A,B) are
2 3
The dimension of the controllable subspace = 5
The input decoupling zeros are the eigenvalues of the matrix AF.
The matrix AF is
-4.0000
The number of finite invariant zeros = 2
The finite invariant zeros are
real part imag part
2.0000
-1.0000
which correspond to the generalized eigenvalues of (lambda*EF - AF).
The number of infinite zeros = 2
2 infinite zero(s) of order 1
The number of right Kronecker indices = 0
The number of left Kronecker indices = 1
The left Kronecker (row) indices of (A,B,C,D) are
2
Multiplicities of infinite eigenvalues
2 2