Purpose
To compute the matrices of the positive feedback controller
| Ak | Bk |
K = |----|----|
| Ck | Dk |
for the shaped plant
| A | B |
G = |---|---|
| C | D |
in the McFarlane/Glover Loop Shaping Design Procedure.
Specification
SUBROUTINE SB10ID( N, M, NP, A, LDA, B, LDB, C, LDC, D, LDD,
$ FACTOR, NK, AK, LDAK, BK, LDBK, CK, LDCK,
$ DK, LDDK, RCOND, IWORK, DWORK, LDWORK, BWORK,
$ INFO )
C .. Scalar Arguments ..
INTEGER INFO, LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD,
$ LDDK, LDWORK, M, N, NK, NP
DOUBLE PRECISION FACTOR
C .. Array Arguments ..
INTEGER IWORK( * )
LOGICAL BWORK( * )
DOUBLE PRECISION A( LDA, * ), AK( LDAK, * ), B( LDB, * ),
$ BK( LDBK, * ), C( LDC, * ), CK( LDCK, * ),
$ D( LDD, * ), DK( LDDK, * ), DWORK( * ),
$ RCOND( 2 )
Arguments
Input/Output Parameters
N (input) INTEGER
The order of the plant. N >= 0.
M (input) INTEGER
The column size of the matrix B. M >= 0.
NP (input) INTEGER
The row size of the matrix C. NP >= 0.
A (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
system state matrix A of the shaped plant.
LDA INTEGER
The leading dimension of the 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
system input matrix B of the shaped plant.
LDB INTEGER
The leading dimension of the array B. LDB >= max(1,N).
C (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading NP-by-N part of this array must contain the
system output matrix C of the shaped plant.
LDC INTEGER
The leading dimension of the array C. LDC >= max(1,NP).
D (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading NP-by-M part of this array must contain the
system matrix D of the shaped plant.
LDD INTEGER
The leading dimension of the array D. LDD >= max(1,NP).
FACTOR (input) DOUBLE PRECISION
= 1 implies that an optimal controller is required;
> 1 implies that a suboptimal controller is required,
achieving a performance FACTOR less than optimal.
FACTOR >= 1.
NK (output) INTEGER
The order of the positive feedback controller. NK <= N.
AK (output) DOUBLE PRECISION array, dimension (LDAK,N)
The leading NK-by-NK part of this array contains the
controller state matrix Ak.
LDAK INTEGER
The leading dimension of the array AK. LDAK >= max(1,N).
BK (output) DOUBLE PRECISION array, dimension (LDBK,NP)
The leading NK-by-NP part of this array contains the
controller input matrix Bk.
LDBK INTEGER
The leading dimension of the array BK. LDBK >= max(1,N).
CK (output) DOUBLE PRECISION array, dimension (LDCK,N)
The leading M-by-NK part of this array contains the
controller output matrix Ck.
LDCK INTEGER
The leading dimension of the array CK. LDCK >= max(1,M).
DK (output) DOUBLE PRECISION array, dimension (LDDK,NP)
The leading M-by-NP part of this array contains the
controller matrix Dk.
LDDK INTEGER
The leading dimension of the array DK. LDDK >= max(1,M).
RCOND (output) DOUBLE PRECISION array, dimension (2)
RCOND(1) contains an estimate of the reciprocal condition
number of the X-Riccati equation;
RCOND(2) contains an estimate of the reciprocal condition
number of the Z-Riccati equation.
Workspace
IWORK INTEGER array, dimension (max(2*N,N*N,M,NP))
DWORK DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal value
of LDWORK.
LDWORK INTEGER
The dimension of the array DWORK.
LDWORK >= 4*N*N + M*M + NP*NP + 2*M*N + N*NP + 4*N +
max( 6*N*N + 5 + max(1,4*N*N+8*N), N*NP + 2*N ).
For good performance, LDWORK must generally be larger.
An upper bound of LDWORK in the above formula is
LDWORK >= 10*N*N + M*M + NP*NP + 2*M*N + 2*N*NP + 4*N +
5 + max(1,4*N*N+8*N).
BWORK LOGICAL array, dimension (2*N)
Error Indicator
INFO INTEGER
= 0: successful exit;
< 0: if INFO = -i, the i-th argument had an illegal
value;
= 1: the X-Riccati equation is not solved successfully;
= 2: the Z-Riccati equation is not solved successfully;
= 3: the iteration to compute eigenvalues or singular
values failed to converge;
= 4: the matrix Ip - D*Dk is singular;
= 5: the matrix Im - Dk*D is singular;
= 6: the closed-loop system is unstable.
Method
The routine implements the formulas given in [1].References
[1] McFarlane, D. and Glover, K.
A loop shaping design procedure using H_infinity synthesis.
IEEE Trans. Automat. Control, vol. AC-37, no. 6, pp. 759-769,
1992.
Numerical Aspects
The accuracy of the results depends on the conditioning of the two Riccati equations solved in the controller design (see the output parameter RCOND).Further Comments
NoneExample
Program Text
* SB10ID 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 = 10, MMAX = 10, PMAX = 10 )
INTEGER LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD, LDDK
PARAMETER ( LDA = NMAX, LDAK = NMAX, LDB = NMAX,
$ LDBK = NMAX, LDC = PMAX, LDCK = MMAX,
$ LDD = PMAX, LDDK = MMAX )
INTEGER LIWORK
PARAMETER ( LIWORK = MAX( 2*NMAX, NMAX*NMAX, MMAX, PMAX ) )
INTEGER LDWORK
PARAMETER ( LDWORK = 4*NMAX*NMAX + MMAX*MMAX + PMAX*PMAX +
$ 2*MMAX*NMAX + NMAX*PMAX + 4*NMAX +
$ MAX( 10*NMAX*NMAX + 8*NMAX + 5,
$ NMAX*PMAX + 2*NMAX ) )
* .. Local Scalars ..
DOUBLE PRECISION FACTOR
INTEGER I, INFO, J, M, N, NK, NP
* .. Local Arrays ..
LOGICAL BWORK(2*NMAX)
INTEGER IWORK(LIWORK)
DOUBLE PRECISION A(LDA,NMAX), AK(LDA,NMAX), B(LDB,MMAX),
$ BK(LDBK,PMAX), C(LDC,NMAX), CK(LDCK,NMAX),
$ D(LDD,MMAX), DK(LDDK,PMAX), DWORK(LDWORK),
$ RCOND( 2 )
* .. External Subroutines ..
EXTERNAL SB10ID
* .. 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, NP
IF ( N.LT.0 .OR. N.GT.NMAX ) THEN
WRITE ( NOUT, FMT = 99990 ) N
ELSE IF ( M.LT.0 .OR. M.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99989 ) M
ELSE IF ( NP.LT.0 .OR. NP.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99988 ) NP
ELSE
READ ( NIN, FMT = * ) ( ( A(I,J), J = 1,N ), I = 1,N )
READ ( NIN, FMT = * ) ( ( B(I,J), J = 1,M ), I = 1,N )
READ ( NIN, FMT = * ) ( ( C(I,J), J = 1,N ), I = 1,NP )
READ ( NIN, FMT = * ) ( ( D(I,J), J = 1,M ), I = 1,NP )
READ ( NIN, FMT = * ) FACTOR
CALL SB10ID( N, M, NP, A, LDA, B, LDB, C, LDC, D, LDD,
$ FACTOR, NK, AK, LDAK, BK, LDBK, CK, LDCK,
$ DK, LDDK, RCOND, IWORK, DWORK, LDWORK,
$ BWORK, INFO )
IF ( INFO.EQ.0 ) THEN
WRITE ( NOUT, FMT = 99997 )
DO 10 I = 1, NK
WRITE ( NOUT, FMT = 99992 ) ( AK(I,J), J = 1,NK )
10 CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, NK
WRITE ( NOUT, FMT = 99992 ) ( BK(I,J), J = 1,NP )
20 CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 30 I = 1, M
WRITE ( NOUT, FMT = 99992 ) ( CK(I,J), J = 1,NK )
30 CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 40 I = 1, M
WRITE ( NOUT, FMT = 99992 ) ( DK(I,J), J = 1,NP )
40 CONTINUE
WRITE( NOUT, FMT = 99993 )
WRITE( NOUT, FMT = 99991 ) ( RCOND(I), I = 1, 2 )
ELSE
WRITE( NOUT, FMT = 99998 ) INFO
END IF
END IF
STOP
*
99999 FORMAT (' SB10ID EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (/' INFO on exit from SB10ID =',I2)
99997 FORMAT (/' The controller state matrix AK is'/)
99996 FORMAT (/' The controller input matrix BK is'/)
99995 FORMAT (/' The controller output matrix CK is'/)
99994 FORMAT (/' The controller matrix DK is'/)
99993 FORMAT (/' The estimated condition numbers are'/)
99992 FORMAT (10(1X,F9.4))
99991 FORMAT ( 2(1X,D12.5))
99990 FORMAT (/' N is out of range.',/' N = ',I5)
99989 FORMAT (/' M is out of range.',/' M = ',I5)
99988 FORMAT (/' NP is out of range.',/' NP = ',I5)
END
Program Data
SB10ID EXAMPLE PROGRAM DATA 6 2 3 -1.0 0.0 4.0 5.0 -3.0 -2.0 -2.0 4.0 -7.0 -2.0 0.0 3.0 -6.0 9.0 -5.0 0.0 2.0 -1.0 -8.0 4.0 7.0 -1.0 -3.0 0.0 2.0 5.0 8.0 -9.0 1.0 -4.0 3.0 -5.0 8.0 0.0 2.0 -6.0 -3.0 -4.0 2.0 0.0 -5.0 -7.0 4.0 -6.0 -3.0 9.0 1.0 -2.0 1.0 -1.0 2.0 -4.0 0.0 -3.0 -3.0 0.0 5.0 -1.0 1.0 1.0 -7.0 5.0 0.0 -8.0 2.0 -2.0 1.0 -2.0 0.0 4.0 5.0 -3.0 1.0Program Results
SB10ID EXAMPLE PROGRAM RESULTS
The controller state matrix AK is
-39.0671 9.9293 22.2322 -27.4113 43.8655
-6.6117 3.0006 11.0878 -11.4130 15.4269
33.6805 -6.6934 -23.9953 14.1438 -33.4358
-32.3191 9.7316 25.4033 -24.0473 42.0517
-44.1655 18.7767 34.8873 -42.4369 50.8437
The controller input matrix BK is
-10.2905 -16.5382 -10.9782
-4.3598 -8.7525 -5.1447
6.5962 1.8975 6.2316
-9.8770 -14.7041 -11.8778
-9.6726 -22.7309 -18.2692
The controller output matrix CK is
-0.6647 -0.0599 -1.0376 0.5619 1.7297
-8.4202 3.9573 7.3094 -7.6283 10.6768
The controller matrix DK is
0.8466 0.4979 -0.6993
-1.2226 -4.8689 -4.5056
The estimated condition numbers are
0.13861D-01 0.90541D-02