## SB10KD

### Positive feedback controller for a discrete-time system

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

Purpose

```  To compute the matrices of the positive feedback controller

| Ak | Bk |
K = |----|----|
| Ck | Dk |

for the shaped plant

| A | B |
G = |---|---|
| C | 0 |

in the Discrete-Time Loop Shaping Design Procedure.

```
Specification
```      SUBROUTINE SB10KD( N, M, NP, A, LDA, B, LDB, C, LDC, FACTOR,
\$                   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, LDDK,
\$                   LDWORK, M, N, 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, * ),
\$                   DK( LDDK, * ), DWORK( * ), RCOND( 4 )

```
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).

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.

AK      (output) DOUBLE PRECISION array, dimension (LDAK,N)
The leading N-by-N 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 N-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-N 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 (4)
RCOND(1) contains an estimate of the reciprocal condition
number of the linear system of equations from
which the solution of the P-Riccati equation is
obtained;
RCOND(2) contains an estimate of the reciprocal condition
number of the linear system of equations from
which the solution of the Q-Riccati equation is
obtained;
RCOND(3) contains an estimate of the reciprocal condition
number of the linear system of equations from
which the solution of the X-Riccati equation is
obtained;
RCOND(4) contains an estimate of the reciprocal condition
number of the matrix Rx + Bx'*X*Bx (see the

```
Workspace
```  IWORK   INTEGER array, dimension (2*max(N,NP+M))

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 >= 15*N*N + 6*N +
max( 14*N+23, 16*N, 2*N+NP+M, 3*(NP+M) ) +
max( N*N, 11*N*NP + 2*M*M + 8*NP*NP + 8*M*N +
4*M*NP + NP ).
For good performance, LDWORK must generally be larger.

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 P-Riccati equation is not solved successfully;
= 2:  the Q-Riccati equation is not solved successfully;
= 3:  the X-Riccati equation is not solved successfully;
= 4:  the iteration to compute eigenvalues failed to
converge;
= 5:  the matrix Rx + Bx'*X*Bx is singular;
= 6:  the closed-loop system is unstable.

```
Method
```  The routine implements the method presented in .

```
References
```   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. For
better conditioning it is advised to take FACTOR > 1.

```
```  None
```
Example

Program Text

```*     SB10KD 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, LDDK
PARAMETER        ( LDA  = NMAX, LDAK = NMAX, LDB  = NMAX,
\$                   LDBK = NMAX, LDC  = PMAX, LDCK = MMAX,
\$                   LDDK = MMAX )
INTEGER          LIWORK
PARAMETER        ( LIWORK = 2*MAX( NMAX, MMAX + PMAX ) )
INTEGER          LDWORK
PARAMETER        ( LDWORK = 15*NMAX*NMAX + 6*NMAX +
\$                            MAX( 14*NMAX + 23, 16*NMAX,
\$                                 2*NMAX+PMAX+MMAX,
\$                                 3*(PMAX+MMAX) ) +
\$                            MAX( NMAX*NMAX,
\$                                 11*NMAX*PMAX + 2*MMAX*MMAX +
\$                                 8*PMAX*PMAX + 8*MMAX*NMAX +
\$                                 4*MMAX*PMAX + PMAX ) )
*     .. Local Scalars ..
DOUBLE PRECISION FACTOR
INTEGER          I, INFO, J, M, N, 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),
\$                 DK(LDDK,PMAX), DWORK(LDWORK), RCOND(4)
*     .. External Subroutines ..
EXTERNAL         SB10KD
*     .. 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 = * ) FACTOR
CALL SB10KD( N, M, NP, A, LDA, B, LDB, C, LDC, FACTOR, 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, N
WRITE ( NOUT, FMT = 99992 ) ( AK(I,J), J = 1,N )
10       CONTINUE
WRITE ( NOUT, FMT = 99996 )
DO 20 I = 1, N
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,N )
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, 4 )
ELSE
WRITE( NOUT, FMT = 99998 ) INFO
END IF
END IF
STOP
*
99999 FORMAT (' SB10KD EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (/' INFO on exit from SB10KD =',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,F8.4))
99991 FORMAT ( 5(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
``` SB10KD EXAMPLE PROGRAM DATA
6     2     2
0.2  0.0  0.3  0.0 -0.3 -0.1
-0.3  0.2 -0.4 -0.3  0.0  0.0
-0.1  0.1 -0.1  0.0  0.0 -0.3
0.1  0.0  0.0 -0.1 -0.1  0.0
0.0  0.3  0.6  0.2  0.1 -0.4
0.2 -0.4  0.0  0.0  0.2 -0.2
-1.0 -2.0
1.0  3.0
-3.0 -4.0
1.0 -2.0
0.0  1.0
1.0  5.0
1.0 -1.0  2.0 -2.0  0.0 -3.0
-3.0  0.0  1.0 -1.0  1.0 -1.0
1.1
```
Program Results
``` SB10KD EXAMPLE PROGRAM RESULTS

The controller state matrix AK is

0.0337   0.0222   0.0858   0.1264  -0.1872   0.1547
0.4457   0.0668  -0.2255  -0.3204  -0.4548  -0.0691
-0.2419  -0.2506  -0.0982  -0.1321  -0.0130  -0.0838
-0.4402   0.3654  -0.0335  -0.2444   0.6366  -0.6469
-0.3623   0.3854   0.4162   0.4502   0.0065   0.1261
-0.0121  -0.4377   0.0604   0.2265  -0.3389   0.4542

The controller input matrix BK is

0.0931  -0.0269
-0.0872   0.1599
0.0956  -0.1469
-0.1728   0.0129
0.2022  -0.1154
0.2419  -0.1737

The controller output matrix CK is

-0.3677   0.2188   0.0403  -0.0854   0.3564  -0.3535
0.1624  -0.0708   0.0058   0.0606  -0.2163   0.1802

The controller matrix DK is

-0.0857  -0.0246
0.0460   0.0074

The estimated condition numbers are

0.11269D-01  0.17596D-01  0.18225D+00  0.75968D-03
```