### Balance & Truncate model reduction for stable systems

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

Purpose

```  To compute a reduced order model (Ar,Br,Cr) for a stable original
state-space representation (A,B,C) by using either the square-root
or the balancing-free square-root Balance & Truncate (B & T)
model reduction method.

```
Specification
```      SUBROUTINE AB09AD( DICO, JOB, EQUIL, ORDSEL, N, M, P, NR, A, LDA,
\$                   B, LDB, C, LDC, HSV, TOL, IWORK, DWORK, LDWORK,
\$                   IWARN, INFO )
C     .. Scalar Arguments ..
CHARACTER         DICO, EQUIL, JOB, ORDSEL
INTEGER           INFO, IWARN, LDA, LDB, LDC, LDWORK, M, N, NR, P
DOUBLE PRECISION  TOL
C     .. Array Arguments ..
INTEGER           IWORK(*)
DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), DWORK(*), HSV(*)

```
Arguments

Mode Parameters

```  DICO    CHARACTER*1
Specifies the type of the original system as follows:
= 'C':  continuous-time system;
= 'D':  discrete-time system.

JOB     CHARACTER*1
Specifies the model reduction approach to be used
as follows:
= 'B':  use the square-root Balance & Truncate method;
= 'N':  use the balancing-free square-root
Balance & Truncate method.

EQUIL   CHARACTER*1
Specifies whether the user wishes to preliminarily
equilibrate the triplet (A,B,C) as follows:
= 'S':  perform equilibration (scaling);
= 'N':  do not perform equilibration.

ORDSEL  CHARACTER*1
Specifies the order selection method as follows:
= 'F':  the resulting order NR is fixed;
= 'A':  the resulting order NR is automatically determined
on basis of the given tolerance TOL.

```
Input/Output Parameters
```  N       (input) INTEGER
The order of the original state-space representation, i.e.
the order of the matrix A.  N >= 0.

M       (input) INTEGER
The number of system inputs.  M >= 0.

P       (input) INTEGER
The number of system outputs.  P >= 0.

NR      (input/output) INTEGER
On entry with ORDSEL = 'F', NR is the desired order of the
resulting reduced order system.  0 <= NR <= N.
On exit, if INFO = 0, NR is the order of the resulting
reduced order model. NR is set as follows:
if ORDSEL = 'F', NR is equal to MIN(NR,NMIN), where NR
is the desired order on entry and NMIN is the order of a
minimal realization of the given system; NMIN is
determined as the number of Hankel singular values greater
than N*EPS*HNORM(A,B,C), where EPS is the machine
precision (see LAPACK Library Routine DLAMCH) and
HNORM(A,B,C) is the Hankel norm of the system (computed
in HSV(1));
if ORDSEL = 'A', NR is equal to the number of Hankel
singular values greater than MAX(TOL,N*EPS*HNORM(A,B,C)).

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.
On exit, if INFO = 0, the leading NR-by-NR part of this
array contains the state dynamics matrix Ar of the reduced
order system.

LDA     INTEGER
The leading dimension of 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 original input/state matrix B.
On exit, if INFO = 0, the leading NR-by-M part of this
array contains the input/state matrix Br of the reduced
order system.

LDB     INTEGER
The leading dimension of array B.  LDB >= MAX(1,N).

C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading P-by-N part of this array must
contain the original state/output matrix C.
On exit, if INFO = 0, the leading P-by-NR part of this
array contains the state/output matrix Cr of the reduced
order system.

LDC     INTEGER
The leading dimension of array C.  LDC >= MAX(1,P).

HSV     (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, it contains the Hankel singular values of
the original system ordered decreasingly. HSV(1) is the
Hankel norm of the system.

```
Tolerances
```  TOL     DOUBLE PRECISION
If ORDSEL = 'A', TOL contains the tolerance for
determining the order of reduced system.
For model reduction, the recommended value is
TOL = c*HNORM(A,B,C), where c is a constant in the
interval [0.00001,0.001], and HNORM(A,B,C) is the
Hankel-norm of the given system (computed in HSV(1)).
For computing a minimal realization, the recommended
value is TOL = N*EPS*HNORM(A,B,C), where EPS is the
machine precision (see LAPACK Library Routine DLAMCH).
This value is used by default if TOL <= 0 on entry.
If ORDSEL = 'F', the value of TOL is ignored.

```
Workspace
```  IWORK   INTEGER array, dimension (LIWORK)
LIWORK = 0, if JOB = 'B';
LIWORK = N, if JOB = 'N'.

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*(2*N+MAX(N,M,P)+5)+N*(N+1)/2).
For optimum performance LDWORK should be larger.

```
Warning Indicator
```  IWARN   INTEGER
= 0:  no warning;
= 1:  with ORDSEL = 'F', the selected order NR is greater
than the order of a minimal realization of the
given system. In this case, the resulting NR is
set automatically to a value corresponding to the
order of a minimal realization of the system.

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value;
= 1:  the reduction of A to the real Schur form failed;
= 2:  the state matrix A is not stable (if DICO = 'C')
or not convergent (if DICO = 'D');
= 3:  the computation of Hankel singular values failed.

```
Method
```  Let be the stable linear system

d[x(t)] = Ax(t) + Bu(t)
y(t)    = Cx(t)                               (1)

where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system. The subroutine AB09AD determines for
the given system (1), the matrices of a reduced order system

d[z(t)] = Ar*z(t) + Br*u(t)
yr(t)   = Cr*z(t)                             (2)

such that

HSV(NR) <= INFNORM(G-Gr) <= 2*[HSV(NR+1) + ... + HSV(N)],

where G and Gr are transfer-function matrices of the systems
(A,B,C) and (Ar,Br,Cr), respectively, and INFNORM(G) is the
infinity-norm of G.

If JOB = 'B', the square-root Balance & Truncate method of 
is used and, for DICO = 'C', the resulting model is balanced.
By setting TOL <= 0, the routine can be used to compute balanced
minimal state-space realizations of stable systems.

If JOB = 'N', the balancing-free square-root version of the
Balance & Truncate method  is used.
By setting TOL <= 0, the routine can be used to compute minimal
state-space realizations of stable systems.

```
References
```   Tombs M.S. and Postlethwaite I.
Truncated balanced realization of stable, non-minimal
state-space systems.
Int. J. Control, Vol. 46, pp. 1319-1330, 1987.

 Varga A.
Efficient minimal realization procedure based on balancing.
Proc. of IMACS/IFAC Symp. MCTS, Lille, France, May 1991,
A. El Moudui, P. Borne, S. G. Tzafestas (Eds.),
Vol. 2, pp. 42-46.

```
Numerical Aspects
```  The implemented methods rely on accuracy enhancing square-root or
balancing-free square-root techniques.
3
The algorithms require less than 30N  floating point operations.

```
```  None
```
Example

Program Text

```*     AB09AD EXAMPLE PROGRAM TEXT
*     Copyright (c) 2002-2017 NICONET e.V.
*
*     .. Parameters ..
INTEGER          NIN, NOUT
PARAMETER        ( NIN = 5, NOUT = 6 )
INTEGER          NMAX, MMAX, PMAX
PARAMETER        ( NMAX = 20, MMAX = 20, PMAX = 20 )
INTEGER          LDA, LDB, LDC
PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX )
INTEGER          LIWORK
PARAMETER        ( LIWORK = NMAX )
INTEGER          LDWORK
PARAMETER        ( LDWORK = NMAX*( 2*NMAX + 5 +
\$                            MAX( NMAX, MMAX, PMAX ) ) +
\$                          ( NMAX*( NMAX + 1 ) )/2 )
*     .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER          I, INFO, IWARN, J, M, N, NR, P
CHARACTER*1      DICO, EQUIL, JOB, ORDSEL
*     .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
\$                 DWORK(LDWORK), HSV(NMAX)
INTEGER          IWORK(LIWORK)
*     .. External Subroutines ..
*     .. Intrinsic Functions ..
INTRINSIC        MAX
*     .. 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, NR, TOL, DICO, JOB, EQUIL, ORDSEL
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) 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 = 99989 ) 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 = 99988 ) P
ELSE
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,P )
*              Find a reduced ssr for (A,B,C).
CALL AB09AD( DICO, JOB, EQUIL, ORDSEL, N, M, P, NR,
\$                      A, LDA, B, LDB, C, LDC, HSV, TOL, IWORK,
\$                      DWORK, LDWORK, IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NR
WRITE ( NOUT, FMT = 99987 )
WRITE ( NOUT, FMT = 99995 ) ( HSV(J), J = 1,N )
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR )
20             CONTINUE
WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40             CONTINUE
WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR )
60             CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB09AD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB09AD = ',I2)
99997 FORMAT (' The order of reduced model = ',I2)
99996 FORMAT (/' The reduced state dynamics matrix Ar is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The reduced input/state matrix Br is ')
99992 FORMAT (/' The reduced state/output matrix Cr is ')
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' P is out of range.',/' P = ',I5)
99987 FORMAT (/' The Hankel singular values HSV are')
END
```
Program Data
``` AB09AD EXAMPLE PROGRAM DATA (Continuous system)
7     2     3     0   1.E-1      C     N     N     A
-0.04165  0.0000  4.9200  -4.9200  0.0000  0.0000  0.0000
-5.2100  -12.500  0.0000   0.0000  0.0000  0.0000  0.0000
0.0000   3.3300 -3.3300   0.0000  0.0000  0.0000  0.0000
0.5450   0.0000  0.0000   0.0000 -0.5450  0.0000  0.0000
0.0000   0.0000  0.0000   4.9200 -0.04165 0.0000  4.9200
0.0000   0.0000  0.0000   0.0000 -5.2100 -12.500  0.0000
0.0000   0.0000  0.0000   0.0000  0.0000  3.3300 -3.3300
0.0000   0.0000
12.500   0.0000
0.0000   0.0000
0.0000   0.0000
0.0000   0.0000
0.0000   12.500
0.0000   0.0000
1.0000   0.0000  0.0000   0.0000  0.0000  0.0000  0.0000
0.0000   0.0000  0.0000   1.0000  0.0000  0.0000  0.0000
0.0000   0.0000  0.0000   0.0000  1.0000  0.0000  0.0000
```
Program Results
``` AB09AD EXAMPLE PROGRAM RESULTS

The order of reduced model =  5

The Hankel singular values HSV are
2.5139   2.0846   1.9178   0.7666   0.5473   0.0253   0.0246

The reduced state dynamics matrix Ar is
1.3451   5.0399   0.0000   0.0000   4.5315
-4.0214  -3.6604   0.0000   0.0000  -0.9056
0.0000   0.0000   0.5124   1.7910   0.0000
0.0000   0.0000  -4.2167  -2.9900   0.0000
1.2402   1.6416   0.0000   0.0000  -0.0586

The reduced input/state matrix Br is
-0.3857   0.3857
-3.1753   3.1753
-0.7447  -0.7447
-3.6872  -3.6872
1.8197  -1.8197

The reduced state/output matrix Cr is
-0.6704   0.1828  -0.6582   0.2222  -0.0104
0.1089   0.4867   0.0000   0.0000   0.8651
0.6704  -0.1828  -0.6582   0.2222   0.0104
```