Purpose
To find a controllable realization for the linear time-invariant
single-input system
dX/dt = A * X + B * U,
Y = C * X,
where A is an N-by-N matrix, B is an N element vector, C is an
P-by-N matrix, and A and B are reduced by this routine to
orthogonal canonical form using (and optionally accumulating)
orthogonal similarity transformations, which are also applied
to C.
Specification
SUBROUTINE TB01ZD( JOBZ, N, P, A, LDA, B, C, LDC, NCONT, Z, LDZ,
$ TAU, TOL, DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBZ
INTEGER INFO, LDA, LDC, LDWORK, LDZ, N, NCONT, P
DOUBLE PRECISION TOL
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(*), C(LDC,*), DWORK(*), TAU(*),
$ Z(LDZ,*)
Arguments
Mode Parameters
JOBZ CHARACTER*1
Indicates whether the user wishes to accumulate in a
matrix Z the orthogonal similarity transformations for
reducing the system, as follows:
= 'N': Do not form Z and do not store the orthogonal
transformations;
= 'F': Do not form Z, but store the orthogonal
transformations in the factored form;
= 'I': Z is initialized to the unit matrix and the
orthogonal transformation matrix Z is returned.
Input/Output Parameters
N (input) INTEGER
The order of the original state-space representation,
i.e. the order of the matrix A. N >= 0.
P (input) INTEGER
The number of system outputs, or of rows of 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 original state dynamics matrix A.
On exit, the leading NCONT-by-NCONT upper Hessenberg
part of this array contains the canonical form of the
state dynamics matrix, given by Z' * A * Z, of a
controllable realization for the original system. The
elements below the first subdiagonal are set to zero.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input/output) DOUBLE PRECISION array, dimension (N)
On entry, the original input/state vector B.
On exit, the leading NCONT elements of this array contain
canonical form of the input/state vector, given by Z' * B,
with all elements but B(1) set to zero.
C (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the output/state matrix C.
On exit, the leading P-by-N part of this array contains
the transformed output/state matrix, given by C * Z, and
the leading P-by-NCONT part contains the output/state
matrix of the controllable realization.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
NCONT (output) INTEGER
The order of the controllable state-space representation.
Z (output) DOUBLE PRECISION array, dimension (LDZ,N)
If JOBZ = 'I', then the leading N-by-N part of this array
contains the matrix of accumulated orthogonal similarity
transformations which reduces the given system to
orthogonal canonical form.
If JOBZ = 'F', the elements below the diagonal, with the
array TAU, represent the orthogonal transformation matrix
as a product of elementary reflectors. The transformation
matrix can then be obtained by calling the LAPACK Library
routine DORGQR.
If JOBZ = 'N', the array Z is not referenced and can be
supplied as a dummy array (i.e. set parameter LDZ = 1 and
declare this array to be Z(1,1) in the calling program).
LDZ INTEGER
The leading dimension of array Z. If JOBZ = 'I' or
JOBZ = 'F', LDZ >= MAX(1,N); if JOBZ = 'N', LDZ >= 1.
TAU (output) DOUBLE PRECISION array, dimension (N)
The elements of TAU contain the scalar factors of the
elementary reflectors used in the reduction of B and A.
Tolerances
TOL DOUBLE PRECISION
The tolerance to be used in determining the
controllability of (A,B). If the user sets TOL > 0, then
the given value of TOL is used as an absolute tolerance;
elements with absolute value less than TOL are considered
neglijible. If the user sets TOL <= 0, then an implicitly
computed, default tolerance, defined by
TOLDEF = N*EPS*MAX( NORM(A), NORM(B) ) is used instead,
where EPS is the machine precision (see LAPACK Library
routine DLAMCH).
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,P).
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.
Method
The Householder matrix which reduces all but the first element of vector B to zero is found and this orthogonal similarity transformation is applied to the matrix A. The resulting A is then reduced to upper Hessenberg form by a sequence of Householder transformations. Finally, the order of the controllable state- space representation (NCONT) is determined by finding the position of the first sub-diagonal element of A which is below an appropriate zero threshold, either TOL or TOLDEF (see parameter TOL); if NORM(B) is smaller than this threshold, NCONT is set to zero, and no computations for reducing the system to orthogonal canonical form are performed. All orthogonal transformations determined in this process are also applied to the matrix C, from the right.References
[1] Konstantinov, M.M., Petkov, P.Hr. and Christov, N.D.
Orthogonal Invariants and Canonical Forms for Linear
Controllable Systems.
Proc. 8th IFAC World Congress, Kyoto, 1, pp. 49-54, 1981.
[2] Hammarling, S.J.
Notes on the use of orthogonal similarity transformations in
control.
NPL Report DITC 8/82, August 1982.
[3] Paige, C.C
Properties of numerical algorithms related to computing
controllability.
IEEE Trans. Auto. Contr., AC-26, pp. 130-138, 1981.
Numerical Aspects
3 The algorithm requires 0(N ) operations and is backward stable.Further Comments
NoneExample
Program Text
* TB01ZD EXAMPLE PROGRAM TEXT
* Copyright (c) 2002-2017 NICONET e.V.
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX, PMAX
PARAMETER ( NMAX = 20, PMAX = 20 )
INTEGER LDA, LDC, LDZ
PARAMETER ( LDA = NMAX, LDC = PMAX, LDZ = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = MAX( NMAX, PMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, J, N, NCONT, P
CHARACTER*1 JOBZ
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(NMAX), C(LDC,NMAX), DWORK(LDWORK),
$ TAU(NMAX), Z(LDZ,NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL TB01ZD, DORGQR
* .. Intrinsic Functions ..
INTRINSIC MAX
* .. Executable Statements ..
*
WRITE ( NOUT, FMT = 99999 )
* Skip the heading in the data file and read in the data.
READ ( NIN, FMT = '()' )
READ ( NIN, FMT = * ) N, P, TOL, JOBZ
IF ( N.LE.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99993 ) N
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( B(I), I = 1,N )
IF ( P.LE.0 .OR. P.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99992 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
* Find a controllable realization for the given system.
CALL TB01ZD( JOBZ, N, P, A, LDA, B, C, LDC, NCONT, Z, LDZ,
$ TAU, TOL, DWORK, LDWORK, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NCONT
DO 20 I = 1, NCONT
WRITE ( NOUT, FMT = 99994 ) ( A(I,J), J = 1,NCONT )
20 CONTINUE
WRITE ( NOUT, FMT = 99996 ) ( B(I), I = 1,NCONT )
WRITE ( NOUT, FMT = 99991 )
DO 30 I = 1, P
WRITE ( NOUT, FMT = 99994 ) ( C(I,J), J = 1,NCONT )
30 CONTINUE
IF ( LSAME( JOBZ, 'F' ) )
$ CALL DORGQR( N, N, N, Z, LDZ, TAU, DWORK, LDWORK,
$ INFO )
IF ( LSAME( JOBZ, 'F' ).OR.LSAME( JOBZ, 'I' ) ) THEN
WRITE ( NOUT, FMT = 99995 )
DO 40 I = 1, N
WRITE ( NOUT, FMT = 99994 ) ( Z(I,J), J = 1,N )
40 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' TB01ZD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from TB01ZD = ',I2)
99997 FORMAT (' The order of the controllable state-space representati',
$ 'on = ',I2,//' The state dynamics matrix A of a controlla',
$ 'ble realization is ')
99996 FORMAT (/' The input/state vector B of a controllable realizatio',
$ 'n is ',/(1X,F8.4))
99995 FORMAT (/' The similarity transformation matrix Z is ')
99994 FORMAT (20(1X,F8.4))
99993 FORMAT (/' N is out of range.',/' N = ',I5)
99992 FORMAT (/' P is out of range.',/' P = ',I5)
99991 FORMAT (/' The output/state matrix C of a controllable realizati',
$ 'on is ')
END
Program Data
TB01ZD EXAMPLE PROGRAM DATA 3 2 0.0 I 1.0 2.0 0.0 4.0 -1.0 0.0 0.0 0.0 1.0 1.0 0.0 1.0 0.0 2.0 1.0 1.0 0.0 0.0Program Results
TB01ZD EXAMPLE PROGRAM RESULTS The order of the controllable state-space representation = 3 The state dynamics matrix A of a controllable realization is 1.0000 1.4142 0.0000 2.8284 -1.0000 2.8284 0.0000 1.4142 1.0000 The input/state vector B of a controllable realization is -1.4142 0.0000 0.0000 The output/state matrix C of a controllable realization is -0.7071 -2.0000 0.7071 -0.7071 0.0000 -0.7071 The similarity transformation matrix Z is -0.7071 0.0000 -0.7071 0.0000 -1.0000 0.0000 -0.7071 0.0000 0.7071