## TB01ZD

### Controllable realization for single-input systems using orthogonal state and input transformations

[Specification] [Arguments] [Method] [References] [Comments] [Example]

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
```   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.

 Hammarling, S.J.
Notes on the use of orthogonal similarity transformations in
control.
NPL Report DITC 8/82, August 1982.

 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.

```
```  None
```
Example

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   0.0
```
Program 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
```