## AB09ED

### Optimal Hankel-norm approximation based model reduction for unstable systems

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

Purpose

```  To compute a reduced order model (Ar,Br,Cr,Dr) for an original
state-space representation (A,B,C,D) by using the optimal
Hankel-norm approximation method in conjunction with square-root
balancing for the ALPHA-stable part of the system.

```
Specification
```      SUBROUTINE AB09ED( DICO, EQUIL, ORDSEL, N, M, P, NR, ALPHA,
\$                   A, LDA, B, LDB, C, LDC, D, LDD, NS, HSV, TOL1,
\$                   TOL2, IWORK, DWORK, LDWORK, IWARN, INFO )
C     .. Scalar Arguments ..
CHARACTER         DICO, EQUIL, ORDSEL
INTEGER           INFO, IWARN, LDA, LDB, LDC, LDD, LDWORK,
\$                  M, N, NR, NS, P
DOUBLE PRECISION  ALPHA, TOL1, TOL2
C     .. Array Arguments ..
INTEGER           IWORK(*)
DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
\$                  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.

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 TOL1.

```
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. For a system with NU ALPHA-unstable
eigenvalues and NS ALPHA-stable eigenvalues (NU+NS = N),
NR is set as follows: if ORDSEL = 'F', NR is equal to
NU+MIN(MAX(0,NR-NU-KR+1),NMIN), where KR is the
multiplicity of the Hankel singular value HSV(NR-NU+1),
NR is the desired order on entry, and NMIN is the order
of a minimal realization of the ALPHA-stable part of the
given system; NMIN is determined as the number of Hankel
singular values greater than NS*EPS*HNORM(As,Bs,Cs), where
EPS is the machine precision (see LAPACK Library Routine
DLAMCH) and HNORM(As,Bs,Cs) is the Hankel norm of the
ALPHA-stable part of the given system (computed in
HSV(1));
if ORDSEL = 'A', NR is the sum of NU and the number of
Hankel singular values greater than
MAX(TOL1,NS*EPS*HNORM(As,Bs,Cs)).

ALPHA   (input) DOUBLE PRECISION
Specifies the ALPHA-stability boundary for the eigenvalues
of the state dynamics matrix A. For a continuous-time
system (DICO = 'C'), ALPHA <= 0 is the boundary value for
the real parts of eigenvalues, while for a discrete-time
system (DICO = 'D'), 0 <= ALPHA <= 1 represents the
boundary value for the moduli of eigenvalues.
The ALPHA-stability domain does not include the boundary.

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 in a real Schur form.
The resulting A has a block-diagonal form with two blocks.
For a system with NU ALPHA-unstable eigenvalues and
NS ALPHA-stable eigenvalues (NU+NS = N), the leading
NU-by-NU block contains the unreduced part of A
corresponding to ALPHA-unstable eigenvalues.
The trailing (NR+NS-N)-by-(NR+NS-N) block contains
the reduced part of A corresponding to ALPHA-stable
eigenvalues.

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).

D       (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the original input/output matrix D.
On exit, if INFO = 0, the leading P-by-M part of this
array contains the input/output matrix Dr of the reduced
order system.

LDD     INTEGER
The leading dimension of array D.  LDD >= MAX(1,P).

NS      (output) INTEGER
The dimension of the ALPHA-stable subsystem.

HSV     (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, the leading NS elements of HSV contain the
Hankel singular values of the ALPHA-stable part of the
original system ordered decreasingly.
HSV(1) is the Hankel norm of the ALPHA-stable subsystem.

```
Tolerances
```  TOL1    DOUBLE PRECISION
If ORDSEL = 'A', TOL1 contains the tolerance for
determining the order of reduced system.
For model reduction, the recommended value is
TOL1 = c*HNORM(As,Bs,Cs), where c is a constant in the
interval [0.00001,0.001], and HNORM(As,Bs,Cs) is the
Hankel-norm of the ALPHA-stable part of the given system
(computed in HSV(1)).
If TOL1 <= 0 on entry, the used default value is
TOL1 = NS*EPS*HNORM(As,Bs,Cs), where NS is the number of
ALPHA-stable eigenvalues of A and EPS is the machine
precision (see LAPACK Library Routine DLAMCH).
This value is appropriate to compute a minimal realization
of the ALPHA-stable part.
If ORDSEL = 'F', the value of TOL1 is ignored.

TOL2    DOUBLE PRECISION
The tolerance for determining the order of a minimal
realization of the ALPHA-stable part of the given system.
The recommended value is TOL2 = NS*EPS*HNORM(As,Bs,Cs).
This value is used by default if TOL2 <= 0 on entry.
If TOL2 > 0, then TOL2 <= TOL1.

```
Workspace
```  IWORK   INTEGER array, dimension (LIWORK)
LIWORK = MAX(1,M),   if DICO = 'C';
LIWORK = MAX(1,N,M), if DICO = 'D'.
On exit, if INFO = 0, IWORK(1) contains NMIN, the order of
the computed minimal realization.

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( LDW1, LDW2 ), where
LDW1 = N*(2*N + MAX(N,M,P) + 5) + N*(N+1)/2,
LDW2 = N*(M+P+2) + 2*M*P + MIN(N,M) +
MAX( 3*M+1, MIN(N,M)+P ).
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 NSMIN, the sum of the order of the
ALPHA-unstable part and the order of a minimal
realization of the ALPHA-stable part of the given
system. In this case, the resulting NR is set equal
to NSMIN.
= 2:  with ORDSEL = 'F', the selected order NR is less
than the order of the ALPHA-unstable part of the
given system. In this case NR is set equal to the
order of the ALPHA-unstable part.

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value;
= 1:  the computation of the ordered real Schur form of A
failed;
= 2:  the separation of the ALPHA-stable/unstable diagonal
blocks failed because of very close eigenvalues;
= 3:  the computed ALPHA-stable part is just stable,
having stable eigenvalues very near to the imaginary
axis (if DICO = 'C') or to the unit circle
(if DICO = 'D');
= 4:  the computation of Hankel singular values failed;
= 5:  the computation of stable projection in the
Hankel-norm approximation algorithm failed;
= 6:  the order of computed stable projection in the
Hankel-norm approximation algorithm differs
from the order of Hankel-norm approximation.

```
Method
```  Let be the following linear system

d[x(t)] = Ax(t) + Bu(t)
y(t)    = Cx(t) + Du(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 AB09ED 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) + Dr*u(t)                       (2)

such that

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

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

The following procedure is used to reduce a given G:

G = G1 + G2

such that G1 = (As,Bs,Cs,D) has only ALPHA-stable poles and
G2 = (Au,Bu,Cu,0) has only ALPHA-unstable poles.

2) Determine G1r, a reduced order approximation of the
ALPHA-stable part G1.

3) Assemble the reduced model Gr as

Gr = G1r + G2.

To reduce the ALPHA-stable part G1, the optimal Hankel-norm
approximation method of [1], based on the square-root
balancing projection formulas of [2], is employed.

```
References
```  [1] Glover, K.
All optimal Hankel norm approximation of linear
multivariable systems and their L-infinity error bounds.
Int. J. Control, Vol. 36, pp. 1145-1193, 1984.

[2] 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.

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

```
```  None
```
Example

Program Text

```*     AB09ED 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, LDD
PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
\$                   LDD = PMAX )
INTEGER          LIWORK
PARAMETER        ( LIWORK = MAX( NMAX, MMAX ) )
INTEGER          LDWORK
PARAMETER        ( LDWORK = MAX( NMAX*( 2*NMAX +
\$                                        MAX( NMAX, MMAX, PMAX ) +
\$                                 5 ) + ( NMAX*( NMAX + 1 ) )/2,
\$                                 NMAX*( MMAX + PMAX + 2 ) +
\$                                 2*MMAX*PMAX + MIN( NMAX, MMAX ) +
\$                                 MAX( 3*MMAX + 1,
\$                                      MIN( NMAX, MMAX ) +
\$                                      PMAX ) ) )
*     .. Local Scalars ..
DOUBLE PRECISION ALPHA, TOL1, TOL2
INTEGER          I, INFO, IWARN, J, M, N, NR, NS, P
CHARACTER*1      DICO, EQUIL, ORDSEL
*     .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
\$                 D(LDD,MMAX), DWORK(LDWORK), HSV(NMAX)
INTEGER          IWORK(LIWORK)
*     .. External Subroutines ..
EXTERNAL         AB09ED
*     .. 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, NR, ALPHA, TOL1, TOL2,
\$                      DICO, 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 )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,P )
*              Find a reduced ssr for (A,B,C,D).
CALL AB09ED( DICO, EQUIL, ORDSEL, N, M, P, NR, ALPHA,
\$                      A, LDA, B, LDB, C, LDC, D, LDD, NS, HSV,
\$                      TOL1, TOL2, 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, NS )
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NR )
20             CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NR
WRITE ( NOUT, FMT = 99995 ) ( B(I,J), J = 1,M )
40             CONTINUE
IF( NR.GT.0 ) WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( C(I,J), J = 1,NR )
60             CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 70 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( D(I,J), J = 1,M )
70             CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' AB09ED EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from AB09ED = ',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 ')
99991 FORMAT (/' The reduced input/output matrix Dr 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 of ALPHA-stable part are')
END
```
Program Data
``` AB09ED EXAMPLE PROGRAM DATA (Continuous system)
7  2   3   0   -0.6D0 1.E-1  1.E-14 C  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
0.0000  0.0000
0.0000  0.0000
0.0000  0.0000

```
Program Results
``` AB09ED EXAMPLE PROGRAM RESULTS

The order of reduced model =  5

The Hankel singular values of ALPHA-stable part are
1.9178   0.8621   0.7666   0.0336   0.0246

The reduced state dynamics matrix Ar is
-0.5181  -1.1084   0.0000   0.0000   0.0000
8.8157  -0.5181   0.0000   0.0000   0.0000
0.0000   0.0000  -1.2769   7.3264   0.0000
0.0000   0.0000  -0.6203  -1.2769   0.0000
0.0000   0.0000   0.0000   0.0000  -1.5496

The reduced input/state matrix Br is
-1.2837   1.2837
-0.7522   0.7522
3.2016   3.2016
-0.7640  -0.7640
1.3415  -1.3415

The reduced state/output matrix Cr is
-0.1380  -0.6445  -0.6247  -2.0857  -0.8964
0.6246   0.0196   0.0000   0.0000   0.6131
0.1380   0.6445  -0.6247  -2.0857   0.8964

The reduced input/output matrix Dr is
0.0168  -0.0168
0.0008  -0.0008
-0.0168   0.0168
```