## AB09CX

### Optimal Hankel-norm approximation based model reduction for stable systems with state matrix in real Schur canonical form

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

Purpose

```  To compute a reduced order model (Ar,Br,Cr,Dr) for a stable
original state-space representation (A,B,C,D) by using the optimal
Hankel-norm approximation method in conjunction with square-root
balancing. The state dynamics matrix A of the original system is
an upper quasi-triangular matrix in real Schur canonical form.

```
Specification
```      SUBROUTINE AB09CX( DICO, ORDSEL, N, M, P, NR, A, LDA, B, LDB,
\$                   C, LDC, D, LDD, HSV, TOL1, TOL2, IWORK,
\$                   DWORK, LDWORK, IWARN, INFO )
C     .. Scalar Arguments ..
CHARACTER         DICO, ORDSEL
INTEGER           INFO, IWARN, LDA, LDB, LDC, LDD, LDWORK,
\$                  M, N, NR, P
DOUBLE PRECISION  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.

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. NR is set as follows:
if ORDSEL = 'F', NR is equal to MIN(MAX(0,NR-KR+1),NMIN),
where KR is the multiplicity of the Hankel singular value
HSV(NR+1), 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(TOL1,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 in a real Schur
canonical form.
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.

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).

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
```  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(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 TOL1 = N*EPS*HNORM(A,B,C), where EPS is the
machine precision (see LAPACK Library Routine DLAMCH).
This value is used by default if TOL1 <= 0 on entry.
If ORDSEL = 'F', the value of TOL1 is ignored.

TOL2    DOUBLE PRECISION
The tolerance for determining the order of a minimal
realization of the given system. The recommended value is
TOL2 = N*EPS*HNORM(A,B,C). 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 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 state matrix A is not stable (if DICO = 'C')
or not convergent (if DICO = 'D');
= 2:  the computation of Hankel singular values failed;
= 3:  the computation of stable projection failed;
= 4:  the order of computed stable projection differs
from the order of Hankel-norm approximation.

```
Method
```  Let be the stable 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 AB09CX 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) <= INFNORM(G-Gr) <= 2*[HSV(NR+1) + ... + HSV(N)],

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 optimal Hankel-norm approximation method of , based on the
square-root balancing projection formulas of , is employed.

```
References
```   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.

 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

```  None
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Program Data
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Program Results
```  None
```