Purpose
To compute, for a given open-loop model (A,B,C,D), and for given state feedback gain F and full observer gain G, such that A+B*F and A+G*C are stable, a reduced order controller model (Ac,Bc,Cc,Dc) using a coprime factorization based controller reduction approach. For reduction, either the square-root or the balancing-free square-root versions of the Balance & Truncate (B&T) or Singular Perturbation Approximation (SPA) model reduction methods are used in conjunction with stable coprime factorization techniques.Specification
SUBROUTINE SB16BD( DICO, JOBD, JOBMR, JOBCF, EQUIL, ORDSEL,
$ N, M, P, NCR, A, LDA, B, LDB, C, LDC, D, LDD,
$ F, LDF, G, LDG, DC, LDDC, HSV, TOL1, TOL2,
$ IWORK, DWORK, LDWORK, IWARN, INFO )
C .. Scalar Arguments ..
CHARACTER DICO, EQUIL, JOBCF, JOBD, JOBMR, ORDSEL
INTEGER INFO, IWARN, LDA, LDB, LDC, LDD, LDDC,
$ LDF, LDG, LDWORK, M, N, NCR, P
DOUBLE PRECISION TOL1, TOL2
C .. Array Arguments ..
INTEGER IWORK(*)
DOUBLE PRECISION A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
$ DC(LDDC,*), DWORK(*), F(LDF,*), G(LDG,*), HSV(*)
Arguments
Mode Parameters
DICO CHARACTER*1
Specifies the type of the open-loop system as follows:
= 'C': continuous-time system;
= 'D': discrete-time system.
JOBD CHARACTER*1
Specifies whether or not a non-zero matrix D appears
in the given state space model:
= 'D': D is present;
= 'Z': D is assumed a zero matrix.
JOBMR CHARACTER*1
Specifies the model reduction approach to be used
as follows:
= 'B': use the square-root B&T method;
= 'F': use the balancing-free square-root B&T method;
= 'S': use the square-root SPA method;
= 'P': use the balancing-free square-root SPA method.
JOBCF CHARACTER*1
Specifies whether left or right coprime factorization is
to be used as follows:
= 'L': use left coprime factorization;
= 'R': use right coprime factorization.
EQUIL CHARACTER*1
Specifies whether the user wishes to perform a
preliminary equilibration before performing
order reduction as follows:
= 'S': perform equilibration (scaling);
= 'N': do not perform equilibration.
ORDSEL CHARACTER*1
Specifies the order selection method as follows:
= 'F': the resulting controller order NCR is fixed;
= 'A': the resulting controller order NCR is
automatically determined on basis of the given
tolerance TOL1.
Input/Output Parameters
N (input) INTEGER
The order of the open-loop state-space representation,
i.e., the order of the matrix A. N >= 0.
N also represents the order of the original state-feedback
controller.
M (input) INTEGER
The number of system inputs. M >= 0.
P (input) INTEGER
The number of system outputs. P >= 0.
NCR (input/output) INTEGER
On entry with ORDSEL = 'F', NCR is the desired order of
the resulting reduced order controller. 0 <= NCR <= N.
On exit, if INFO = 0, NCR is the order of the resulting
reduced order controller. NCR is set as follows:
if ORDSEL = 'F', NCR is equal to MIN(NCR,NMIN), where NCR
is the desired order on entry, and NMIN is the order of a
minimal realization of an extended system Ge (see METHOD);
NMIN is determined as the number of
Hankel singular values greater than N*EPS*HNORM(Ge),
where EPS is the machine precision (see LAPACK Library
Routine DLAMCH) and HNORM(Ge) is the Hankel norm of the
extended system (computed in HSV(1));
if ORDSEL = 'A', NCR is equal to the number of Hankel
singular values greater than MAX(TOL1,N*EPS*HNORM(Ge)).
A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the original state dynamics matrix A.
On exit, if INFO = 0, the leading NCR-by-NCR part of this
array contains the state dynamics matrix Ac of the reduced
controller.
LDA INTEGER
The leading dimension of array A. LDA >= MAX(1,N).
B (input) DOUBLE PRECISION array, dimension (LDB,M)
The leading N-by-M part of this array must
contain the original input/state matrix B.
LDB INTEGER
The leading dimension of array B. LDB >= MAX(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading P-by-N part of this array must
contain the original state/output matrix C.
LDC INTEGER
The leading dimension of array C. LDC >= MAX(1,P).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
If JOBD = 'D', the leading P-by-M part of this
array must contain the system direct input/output
transmission matrix D.
The array D is not referenced if JOBD = 'Z'.
LDD INTEGER
The leading dimension of array D.
LDD >= MAX(1,P), if JOBD = 'D';
LDD >= 1, if JOBD = 'Z'.
F (input/output) DOUBLE PRECISION array, dimension (LDF,N)
On entry, the leading M-by-N part of this array must
contain a stabilizing state feedback matrix.
On exit, if INFO = 0, the leading M-by-NCR part of this
array contains the state/output matrix Cc of the reduced
controller.
LDF INTEGER
The leading dimension of array F. LDF >= MAX(1,M).
G (input/output) DOUBLE PRECISION array, dimension (LDG,P)
On entry, the leading N-by-P part of this array must
contain a stabilizing observer gain matrix.
On exit, if INFO = 0, the leading NCR-by-P part of this
array contains the input/state matrix Bc of the reduced
controller.
LDG INTEGER
The leading dimension of array G. LDG >= MAX(1,N).
DC (output) DOUBLE PRECISION array, dimension (LDDC,P)
If INFO = 0, the leading M-by-P part of this array
contains the input/output matrix Dc of the reduced
controller.
LDDC INTEGER
The leading dimension of array DC. LDDC >= MAX(1,M).
HSV (output) DOUBLE PRECISION array, dimension (N)
If INFO = 0, it contains the N Hankel singular values
of the extended system ordered decreasingly (see METHOD).
Tolerances
TOL1 DOUBLE PRECISION
If ORDSEL = 'A', TOL1 contains the tolerance for
determining the order of the reduced extended system.
For model reduction, the recommended value is
TOL1 = c*HNORM(Ge), where c is a constant in the
interval [0.00001,0.001], and HNORM(Ge) is the
Hankel norm of the extended system (computed in HSV(1)).
The value TOL1 = N*EPS*HNORM(Ge) is used by default if
TOL1 <= 0 on entry, 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 coprime factorization controller
(see METHOD). The recommended value is
TOL2 = N*EPS*HNORM(Ge) (see METHOD).
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 (LIWORK)
LIWORK = 0, if ORDSEL = 'F' and NCR = N.
Otherwise,
LIWORK = MAX(PM,M), if JOBCF = 'L',
LIWORK = MAX(PM,P), if JOBCF = 'R', where
PM = 0, if JOBMR = 'B',
PM = N, if JOBMR = 'F',
PM = MAX(1,2*N), if JOBMR = 'S' or 'P'.
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 >= P*N, if ORDSEL = 'F' and NCR = N. Otherwise,
LDWORK >= (N+M)*(M+P) + MAX(LWR,4*M), if JOBCF = 'L',
LDWORK >= (N+P)*(M+P) + MAX(LWR,4*P), if JOBCF = 'R',
where LWR = 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 NCR is
greater than the order of a minimal
realization of the controller.
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the reduction of A+G*C to a real Schur form
failed;
= 2: the matrix A+G*C is not stable (if DICO = 'C'),
or not convergent (if DICO = 'D');
= 3: the computation of Hankel singular values failed;
= 4: the reduction of A+B*F to a real Schur form
failed;
= 5: the matrix A+B*F is not stable (if DICO = 'C'),
or not convergent (if DICO = 'D').
Method
Let be the 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, and let Go(d) be the open-loop
transfer-function matrix
-1
Go(d) = C*(d*I-A) *B + D .
Let F and G be the state feedback and observer gain matrices,
respectively, chosen so that A+B*F and A+G*C are stable matrices.
The controller has a transfer-function matrix K(d) given by
-1
K(d) = F*(d*I-A-B*F-G*C-G*D*F) *G .
The closed-loop transfer-function matrix is given by
-1
Gcl(d) = Go(d)(I+K(d)Go(d)) .
K(d) can be expressed as a left coprime factorization (LCF),
-1
K(d) = M_left(d) *N_left(d) ,
or as a right coprime factorization (RCF),
-1
K(d) = N_right(d)*M_right(d) ,
where M_left(d), N_left(d), N_right(d), and M_right(d) are
stable transfer-function matrices.
The subroutine SB16BD determines the matrices of a reduced
controller
d[z(t)] = Ac*z(t) + Bc*y(t)
u(t) = Cc*z(t) + Dc*y(t), (2)
with the transfer-function matrix Kr as follows:
(1) If JOBCF = 'L', the extended system
Ge(d) = [ N_left(d) M_left(d) ] is reduced to
Ger(d) = [ N_leftr(d) M_leftr(d) ] by using either the
B&T or SPA methods. The reduced order controller Kr(d)
is computed as
-1
Kr(d) = M_leftr(d) *N_leftr(d) ;
(2) If JOBCF = 'R', the extended system
Ge(d) = [ N_right(d) ] is reduced to
[ M_right(d) ]
Ger(d) = [ N_rightr(d) ] by using either the
[ M_rightr(d) ]
B&T or SPA methods. The reduced order controller Kr(d)
is computed as
-1
Kr(d) = N_rightr(d)* M_rightr(d) .
If ORDSEL = 'A', the order of the controller is determined by
computing the number of Hankel singular values greater than
the given tolerance TOL1. The Hankel singular values are
the square roots of the eigenvalues of the product of
the controllability and observability Grammians of the
extended system Ge.
If JOBMR = 'B', the square-root B&T method of [1] is used.
If JOBMR = 'F', the balancing-free square-root version of the
B&T method [1] is used.
If JOBMR = 'S', the square-root version of the SPA method [2,3]
is used.
If JOBMR = 'P', the balancing-free square-root version of the
SPA method [2,3] is used.
References
[1] 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.
[2] 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, 1991.
[3] Varga, A.
Coprime factors model reduction method based on square-root
balancing-free techniques.
System Analysis, Modelling and Simulation, Vol. 11,
pp. 303-311, 1993.
[4] Liu, Y., Anderson, B.D.O. and Ly, O.L.
Coprime factorization controller reduction with Bezout
identity induced frequency weighting.
Automatica, vol. 26, pp. 233-249, 1990.
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.
Further Comments
NoneExample
Program Text
* SB16BD 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, LDDC, LDF, LDG
PARAMETER ( LDA = NMAX, LDB = NMAX, LDC = PMAX,
$ LDD = PMAX, LDDC = MMAX, LDF = MMAX, LDG = NMAX
$ )
INTEGER LDWORK, LIWORK, MAXMP, MPMAX
PARAMETER ( LIWORK = 2*NMAX, MAXMP = MAX( MMAX, PMAX ),
$ MPMAX = MMAX + PMAX )
PARAMETER ( LDWORK = ( NMAX + MAXMP )*MPMAX +
$ MAX ( NMAX*( 2*NMAX +
$ MAX( NMAX, MPMAX ) + 5 )
$ + ( NMAX*( NMAX + 1 ) )/2,
$ 4*MAXMP ) )
CHARACTER DICO, EQUIL, JOBCF, JOBD, JOBMR, ORDSEL
INTEGER I, INFO, IWARN, J, M, N, NCR, P
DOUBLE PRECISION TOL1, TOL2
* .. Local Arrays ..
DOUBLE PRECISION A(LDA,NMAX), B(LDB,MMAX), C(LDC,NMAX),
$ D(LDD,MMAX), DC(LDDC,PMAX), DWORK(LDWORK),
$ F(LDF,NMAX), G(LDG,PMAX), HSV(NMAX)
INTEGER IWORK(LIWORK)
* .. External Subroutines ..
EXTERNAL SB16BD
* .. 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, NCR, TOL1, TOL2,
$ DICO, JOBD, JOBMR, JOBCF, 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 )
READ ( NIN, FMT = * ) ( ( F(I,J), J = 1,N ), I = 1,M )
READ ( NIN, FMT = * ) ( ( G(I,J), J = 1,P ), I = 1,N )
* Find a reduced ssr for (A,B,C,D).
CALL SB16BD( DICO, JOBD, JOBMR, JOBCF, EQUIL, ORDSEL, N,
$ M, P, NCR, A, LDA, B, LDB, C, LDC, D, LDD,
$ F, LDF, G, LDG, DC, LDDC, HSV, TOL1, TOL2,
$ IWORK, DWORK, LDWORK, IWARN, INFO )
*
IF ( INFO.NE.0 ) THEN
WRITE ( NOUT, FMT = 99998 ) INFO
ELSE
WRITE ( NOUT, FMT = 99997 ) NCR
WRITE ( NOUT, FMT = 99987 )
WRITE ( NOUT, FMT = 99995 ) ( HSV(J), J = 1,N )
IF( NCR.GT.0 ) WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NCR
WRITE ( NOUT, FMT = 99995 ) ( A(I,J), J = 1,NCR )
20 CONTINUE
IF( NCR.GT.0 ) WRITE ( NOUT, FMT = 99993 )
DO 40 I = 1, NCR
WRITE ( NOUT, FMT = 99995 ) ( G(I,J), J = 1,P )
40 CONTINUE
IF( NCR.GT.0 ) WRITE ( NOUT, FMT = 99992 )
DO 60 I = 1, M
WRITE ( NOUT, FMT = 99995 ) ( F(I,J), J = 1,NCR )
60 CONTINUE
WRITE ( NOUT, FMT = 99991 )
DO 80 I = 1, P
WRITE ( NOUT, FMT = 99995 ) ( DC(I,J), J = 1,M )
80 CONTINUE
END IF
END IF
END IF
END IF
STOP
*
99999 FORMAT (' SB16BD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (' INFO on exit from SB16BD = ',I2)
99997 FORMAT (' The order of reduced controller = ',I2)
99996 FORMAT (/' The reduced controller state dynamics matrix Ac is ')
99995 FORMAT (20(1X,F8.4))
99993 FORMAT (/' The reduced controller input/state matrix Bc is ')
99992 FORMAT (/' The reduced controller state/output matrix Cc is ')
99991 FORMAT (/' The reduced controller input/output matrix Dc 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 extended system are:')
END
Program Data
SB16BD EXAMPLE PROGRAM DATA (Continuous system)
8 1 1 4 0.1E0 0.0 C D F L S F
0 1.0000 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0 0 -0.0150 0.7650 0 0 0 0
0 0 -0.7650 -0.0150 0 0 0 0
0 0 0 0 -0.0280 1.4100 0 0
0 0 0 0 -1.4100 -0.0280 0 0
0 0 0 0 0 0 -0.0400 1.850
0 0 0 0 0 0 -1.8500 -0.040
0.0260
-0.2510
0.0330
-0.8860
-4.0170
0.1450
3.6040
0.2800
-.996 -.105 0.261 .009 -.001 -.043 0.002 -0.026
0.0
4.4721e-002 6.6105e-001 4.6986e-003 3.6014e-001 1.0325e-001 -3.7541e-002 -4.2685e-002 3.2873e-002
4.1089e-001
8.6846e-002
3.8523e-004
-3.6194e-003
-8.8037e-003
8.4205e-003
1.2349e-003
4.2632e-003
Program Results
SB16BD EXAMPLE PROGRAM RESULTS The order of reduced controller = 4 The Hankel singular values of extended system are: 4.9078 4.8745 3.8455 3.7811 1.2289 1.1785 0.5176 0.1148 The reduced controller state dynamics matrix Ac is 0.5946 -0.7336 0.1914 -0.3368 0.5960 -0.0184 -0.1088 0.0207 1.2253 0.2043 0.1009 -1.4948 -0.0330 -0.0243 1.3440 0.0035 The reduced controller input/state matrix Bc is 0.0015 -0.0202 0.0159 -0.0544 The reduced controller state/output matrix Cc is 0.3534 0.0274 0.0337 -0.0320 The reduced controller input/output matrix Dc is 0.0000