Purpose
To compute a reduced order model (Ar,Br,Cr,Dr) for an original
stable state-space representation (A,B,C,D) by using the
stochastic balancing approach in conjunction with the square-root
or the balancing-free square-root Balance & Truncate (B&T) or
Singular Perturbation Approximation (SPA) model reduction methods.
The state dynamics matrix A of the original system is an upper
quasi-triangular matrix in real Schur canonical form and D must be
full row rank.
For the B&T approach, the matrices of the reduced order system
are computed using the truncation formulas:
Ar = TI * A * T , Br = TI * B , Cr = C * T . (1)
For the SPA approach, the matrices of a minimal realization
(Am,Bm,Cm) are computed using the truncation formulas:
Am = TI * A * T , Bm = TI * B , Cm = C * T . (2)
Am, Bm, Cm and D serve further for computing the SPA of the given
system.
Specification
SUBROUTINE AB09HX( 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, BWORK, 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,*)
LOGICAL BWORK(*)
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;
= 'F': use the balancing-free square-root
Balance & Truncate method;
= 'S': use the square-root Singular Perturbation
Approximation method;
= 'P': use the balancing-free square-root
Singular Perturbation Approximation 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. M >= 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, where EPS is the machine precision
(see LAPACK Library Routine DLAMCH);
if ORDSEL = 'A', NR is equal to the number of Hankel
singular values greater than MAX(TOL1,N*EPS).
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,
ordered decreasingly, of the phase system. All singular
values are less than or equal to 1.
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), for
the B&T approach, or in (2), for the SPA approach.
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), for
the B&T approach, or in (2), for the SPA approach.
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 lies in the
interval [0.00001,0.001].
If TOL1 <= 0 on entry, the used default value is
TOL1 = N*EPS, where EPS is the machine
precision (see LAPACK Library Routine DLAMCH).
If ORDSEL = 'F', the value of TOL1 is ignored.
TOL2 DOUBLE PRECISION
The tolerance for determining the order of a minimal
realization of the phase system (see METHOD) corresponding
to the given system.
The recommended value is TOL2 = N*EPS.
This value is used by default if TOL2 <= 0 on entry.
If TOL2 > 0 and ORDSEL = 'A', 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 and DWORK(2) contains RCOND, the reciprocal
condition number of the U11 matrix from the expression
used to compute the solution X = U21*inv(U11) of the
Riccati equation for spectral factorization.
A small value RCOND indicates possible ill-conditioning
of the respective Riccati equation.
LDWORK INTEGER
The length of the array DWORK.
LDWORK >= MAX( 2, N*(MAX(N,M,P)+5),
2*N*P+MAX(P*(M+2),10*N*(N+1) ) ).
For optimum performance LDWORK should be larger.
BWORK LOGICAL array, dimension 2*N
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'), or it is not in
a real Schur form;
= 2: the reduction of Hamiltonian matrix to real
Schur form failed;
= 3: the reordering of the real Schur form of the
Hamiltonian matrix failed;
= 4: the Hamiltonian matrix has less than N stable
eigenvalues;
= 5: the coefficient matrix U11 in the linear system
X*U11 = U21, used to determine X, is singular to
working precision;
= 6: the feedthrough matrix D has not a full row rank P;
= 7: 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), (3)
where d[x(t)] is dx(t)/dt for a continuous-time system and x(t+1)
for a discrete-time system. The subroutine AB09HX determines for
the given system (3), the matrices of a reduced NR-rder system
d[z(t)] = Ar*z(t) + Br*u(t)
yr(t) = Cr*z(t) + Dr*u(t), (4)
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 square-root stochastic Balance & Truncate
method of [1] is used and the resulting model is balanced.
If JOB = 'F', the balancing-free square-root version of the
stochastic Balance & Truncate method [1] is used.
If JOB = 'S', the stochastic balancing method, in conjunction
with the square-root version of the Singular Perturbation
Approximation method [2,3] is used.
If JOB = 'P', the stochastic balancing method, in conjunction
with the balancing-free square-root version of the Singular
Perturbation Approximation method [2,3] is used.
By setting TOL1 = TOL2, the routine can be also used to compute
Balance & Truncate approximations.
References
[1] Varga A. and Fasol K.H.
A new square-root balancing-free stochastic truncation
model reduction algorithm.
Proc. of 12th IFAC World Congress, Sydney, 1993.
[2] Liu Y. and Anderson B.D.O.
Singular Perturbation Approximation of balanced systems.
Int. J. Control, Vol. 50, pp. 1379-1405, 1989.
[3] 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 method relies on accuracy enhancing square-root
or balancing-free square-root methods. The effectiveness of the
accuracy enhancing technique depends on the accuracy of the
solution of a Riccati equation. Ill-conditioned Riccati solution
typically results when D is nearly rank deficient.
3
The algorithm requires about 100N floating point operations.
Further Comments
NoneExample
Program Text
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NoneProgram Results
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