## AB09BX

### Singular Perturbation 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 either the
square-root or the balancing-free square-root
Singular Perturbation Approximation (SPA) model reduction method.
The state dynamics matrix A of the original system is an upper
quasi-triangular matrix in real Schur canonical form. The matrices
of a minimal realization are computed using the truncation
formulas:

Am = TI * A * T ,  Bm = TI * B ,  Cm = C * T .      (1)

Am, Bm, Cm and D serve further for computing the SPA of the given
system.

```
Specification
```      SUBROUTINE AB09BX( DICO, JOB, ORDSEL, N, M, P, NR, A, LDA, B, LDB,
\$                   C, LDC, D, LDD, HSV, T, LDT, TI, LDTI, TOL1,
\$                   TOL2, IWORK, DWORK, LDWORK, IWARN, INFO )
C     .. Scalar Arguments ..
CHARACTER         DICO, JOB, ORDSEL
INTEGER           INFO, IWARN, LDA, LDB, LDC, LDD, LDT, LDTI,
\$                  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(*), T(LDT,*), TI(LDTI,*)

```
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 SPA method;
= 'N':  use the balancing-free square-root SPA method.

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(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(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.

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.

T       (output) DOUBLE PRECISION array, dimension (LDT,N)
If INFO = 0 and NR > 0, the leading N-by-NR part of this
array contains the right truncation matrix T in (1).

LDT     INTEGER
The leading dimension of array T.  LDT >= MAX(1,N).

TI      (output) DOUBLE PRECISION array, dimension (LDTI,N)
If INFO = 0 and NR > 0, the leading NR-by-N part of this
array contains the left truncation matrix TI in (1).

LDTI    INTEGER
The leading dimension of array TI.  LDTI >= MAX(1,N).

```
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 (MAX(1,2*N))
On exit with INFO = 0, IWORK(1) contains the order of the
minimal realization of the system.

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*(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 state matrix A is not stable (if DICO = 'C')
or not convergent (if DICO = 'D');
= 2:  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) + Du(t)                              (2)

where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system. The subroutine AB09BX determines for
the given system (1), the matrices of a reduced NR order system

d[z(t)] = Ar*z(t) + Br*u(t)
yr(t)   = Cr*z(t) + Dr*u(t)                          (3)

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.

If JOB = 'B', the balancing-based square-root SPA method of 
is used and the resulting model is balanced.

If JOB = 'N', the balancing-free square-root SPA method of 
is used.
By setting TOL1 = TOL2, the routine can be also used to compute
Balance & Truncate approximations.

```
References
```   Liu Y. and Anderson B.D.O.
Singular Perturbation Approximation of Balanced Systems,
Int. J. Control, Vol. 50, pp. 1379-1405, 1989.

 Varga A.
Balancing-free square-root algorithm for computing singular
perturbation approximations.
Proc. 30-th IEEE CDC,  Brighton, Dec. 11-13, 1991,
Vol. 2, pp. 1062-1065.

```
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.

```
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Example

Program Text

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Program Data
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Program Results
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