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<HTML>
<HEAD><TITLE>AB01MD - SLICOT Library Routine Documentation</TITLE>
</HEAD>
<BODY>
<H2><A Name="AB01MD">AB01MD</A></H2>
<H3>
Controllable realization for single-input systems using orthogonal state and input transformations
</H3>
<A HREF ="#Specification"><B>[Specification]</B></A>
<A HREF ="#Arguments"><B>[Arguments]</B></A>
<A HREF ="#Method"><B>[Method]</B></A>
<A HREF ="#References"><B>[References]</B></A>
<A HREF ="#Comments"><B>[Comments]</B></A>
<A HREF ="#Example"><B>[Example]</B></A>
<P>
<B><FONT SIZE="+1">Purpose</FONT></B>
<PRE>
To find a controllable realization for the linear time-invariant
single-input system
dX/dt = A * X + B * U,
where A is an N-by-N matrix and B is an N element vector which
are reduced by this routine to orthogonal canonical form using
(and optionally accumulating) orthogonal similarity
transformations.
</PRE>
<A name="Specification"><B><FONT SIZE="+1">Specification</FONT></B></A>
<PRE>
SUBROUTINE AB01MD( JOBZ, N, A, LDA, B, NCONT, Z, LDZ, TAU, TOL,
$ DWORK, LDWORK, INFO )
C .. Scalar Arguments ..
CHARACTER JOBZ
INTEGER INFO, LDA, LDZ, LDWORK, N, NCONT
DOUBLE PRECISION TOL
C .. Array Arguments ..
DOUBLE PRECISION A(LDA,*), B(*), DWORK(*), TAU(*), Z(LDZ,*)
</PRE>
<A name="Arguments"><B><FONT SIZE="+1">Arguments</FONT></B></A>
<P>
<B>Mode Parameters</B>
<PRE>
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.
</PRE>
<B>Input/Output Parameters</B>
<PRE>
N (input) INTEGER
The order of the original state-space representation,
i.e. the order of the matrix A. N >= 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.
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.
</PRE>
<B>Tolerances</B>
<PRE>
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).
</PRE>
<B>Workspace</B>
<PRE>
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).
For optimum performance LDWORK should be larger.
</PRE>
<B>Error Indicator</B>
<PRE>
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value.
</PRE>
<A name="Method"><B><FONT SIZE="+1">Method</FONT></B></A>
<PRE>
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.
</PRE>
<A name="References"><B><FONT SIZE="+1">References</FONT></B></A>
<PRE>
[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.
</PRE>
<A name="Numerical Aspects"><B><FONT SIZE="+1">Numerical Aspects</FONT></B></A>
<PRE> 3
The algorithm requires 0(N ) operations and is backward stable.
</PRE>
<A name="Comments"><B><FONT SIZE="+1">Further Comments</FONT></B></A>
<PRE>
None
</PRE>
<A name="Example"><B><FONT SIZE="+1">Example</FONT></B></A>
<P>
<B>Program Text</B>
<PRE>
* AB01MD EXAMPLE PROGRAM TEXT
*
* .. Parameters ..
INTEGER NIN, NOUT
PARAMETER ( NIN = 5, NOUT = 6 )
INTEGER NMAX
PARAMETER ( NMAX = 20 )
INTEGER LDA, LDZ
PARAMETER ( LDA = NMAX, LDZ = NMAX )
INTEGER LDWORK
PARAMETER ( LDWORK = NMAX )
* .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER I, INFO, J, N, NCONT
CHARACTER*1 JOBZ
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(NMAX), DWORK(LDWORK), TAU(NMAX),
$ Z(LDZ,NMAX)
* .. External Functions ..
LOGICAL LSAME
EXTERNAL LSAME
* .. External Subroutines ..
EXTERNAL AB01MD, DORGQR
* .. 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, 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 )
* Find a controllable realization for the given system.
CALL AB01MD( JOBZ, N, A, LDA, B, 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 )
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
STOP
*
99999 FORMAT (' AB01MD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB01MD = ',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)
END
</PRE>
<B>Program Data</B>
<PRE>
AB01MD EXAMPLE PROGRAM DATA
3 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
</PRE>
<B>Program Results</B>
<PRE>
AB01MD 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 similarity transformation matrix Z is
-0.7071 0.0000 -0.7071
0.0000 -1.0000 0.0000
-0.7071 0.0000 0.7071
</PRE>
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