## SB10ED

### H2 optimal state controller for a discrete-time system

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

Purpose

```  To compute the matrices of the H2 optimal n-state controller

| AK | BK |
K = |----|----|
| CK | DK |

for the discrete-time system

| A  | B1  B2  |   | A | B |
P = |----|---------| = |---|---| ,
| C1 |  0  D12 |   | C | D |
| C2 | D21 D22 |

where B2 has as column size the number of control inputs (NCON)
and C2 has as row size the number of measurements (NMEAS) being
provided to the controller.

It is assumed that

(A1) (A,B2) is stabilizable and (C2,A) is detectable,

(A2) D12 is full column rank and D21 is full row rank,

j*Theta
(A3) | A-e       *I  B2  | has full column rank for all
|    C1         D12 |

0 <= Theta < 2*Pi ,

j*Theta
(A4) | A-e       *I  B1  |  has full row rank for all
|    C2         D21 |

0 <= Theta < 2*Pi .

```
Specification
```      SUBROUTINE SB10ED( N, M, NP, NCON, NMEAS, A, LDA, B, LDB, C, LDC,
\$                   D, LDD, AK, LDAK, BK, LDBK, CK, LDCK, DK, LDDK,
\$                   RCOND, TOL, IWORK, DWORK, LDWORK, BWORK, INFO )
C     .. Scalar Arguments ..
INTEGER            INFO, LDA, LDAK, LDB, LDBK, LDC, LDCK, LDD,
\$                   LDDK, LDWORK, M, N, NCON, NMEAS, NP
DOUBLE PRECISION   TOL
C     .. Array Arguments ..
INTEGER            IWORK( * )
DOUBLE PRECISION   A( LDA, * ), AK( LDAK, * ), B( LDB, * ),
\$                   BK( LDBK, * ), C( LDC, * ), CK( LDCK, * ),
\$                   D( LDD, * ), DK( LDDK, * ), DWORK( * ),
\$                   RCOND( * )
LOGICAL            BWORK( * )

```
Arguments

Input/Output Parameters

```  N       (input) INTEGER
The order of the system.  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.

NCON    (input) INTEGER
The number of control inputs (M2).  M >= NCON >= 0,
NP-NMEAS >= NCON.

NMEAS   (input) INTEGER
The number of measurements (NP2).  NP >= NMEAS >= 0,
M-NCON >= NMEAS.

A       (input/worksp.) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
system state matrix A.
This array is modified internally, but it is restored on
exit.

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.

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.

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 input/output matrix D.

LDD     INTEGER
The leading dimension of the array D.  LDD >= max(1,NP).

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,NMEAS)
The leading N-by-NMEAS 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 NCON-by-N part of this array contains the
controller output matrix CK.

LDCK    INTEGER
The leading dimension of the array CK.
LDCK >= max(1,NCON).

DK      (output) DOUBLE PRECISION array, dimension (LDDK,NMEAS)
The leading NCON-by-NMEAS part of this array contains the
controller input/output matrix DK.

LDDK    INTEGER
The leading dimension of the array DK.
LDDK >= max(1,NCON).

RCOND   (output) DOUBLE PRECISION array, dimension (7)
RCOND contains estimates the reciprocal condition
numbers of the matrices which are to be inverted and the
reciprocal condition numbers of the Riccati equations
which have to be solved during the computation of the
controller. (See the description of the algorithm in .)
RCOND(1) contains the reciprocal condition number of the
control transformation matrix TU;
RCOND(2) contains the reciprocal condition number of the
measurement transformation matrix TY;
RCOND(3) contains the reciprocal condition number of the
matrix Im2 + B2'*X2*B2;
RCOND(4) contains the reciprocal condition number of the
matrix Ip2 + C2*Y2*C2';
RCOND(5) contains the reciprocal condition number of the
X-Riccati equation;
RCOND(6) contains the reciprocal condition number of the
Y-Riccati equation;
RCOND(7) contains the reciprocal condition number of the
matrix Im2 + DKHAT*D22 .

```
Tolerances
```  TOL     DOUBLE PRECISION
Tolerance used for controlling the accuracy of the
transformations applied for diagonalizing D12 and D21,
and for checking the nonsingularity of the matrices to be
inverted. If TOL <= 0, then a default value equal to
sqrt(EPS) is used, where EPS is the relative machine
precision.

```
Workspace
```  IWORK   INTEGER array, dimension (max(2*M2,2*N,N*N,NP2))

DWORK   DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) contains the optimal
LDWORK.

LDWORK  INTEGER
The dimension of the array DWORK.
LDWORK >= N*M + NP*(N+M) + M2*M2 + NP2*NP2 +
max(1,LW1,LW2,LW3,LW4,LW5,LW6), where
LW1 = (N+NP1+1)*(N+M2) + max(3*(N+M2)+N+NP1,5*(N+M2)),
LW2 = (N+NP2)*(N+M1+1) + max(3*(N+NP2)+N+M1,5*(N+NP2)),
LW3 = M2 + NP1*NP1 + max(NP1*max(N,M1),3*M2+NP1,5*M2),
LW4 = NP2 + M1*M1 + max(max(N,NP1)*M1,3*NP2+M1,5*NP2),
LW5 = 2*N*N+max(1,14*N*N+6*N+max(14*N+23,16*N),M2*(N+M2+
max(3,M1)),NP2*(N+NP2+3)),
LW6 = max(N*M2,N*NP2,M2*NP2,M2*M2+4*M2),
with M1 = M - M2 and NP1 = NP - NP2.
For good performance, LDWORK must generally be larger.
Denoting Q = max(M1,M2,NP1,NP2), an upper bound is
2*Q*(3*Q+2*N)+max(1,(N+Q)*(N+Q+6),Q*(Q+max(N,Q,5)+1),
2*N*N+max(1,14*N*N+6*N+max(14*N+23,16*N),
Q*(N+Q+max(Q,3)))).

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;
j*Theta
= 1:  if the matrix | A-e       *I  B2  | had not full
|      C1       D12 |
column rank in respect to the tolerance EPS;
j*Theta
= 2:  if the matrix | A-e       *I  B1  |  had not full
|      C2       D21 |
row rank in respect to the tolerance EPS;
= 3:  if the matrix D12 had not full column rank in
respect to the tolerance TOL;
= 4:  if the matrix D21 had not full row rank in respect
to the tolerance TOL;
= 5:  if the singular value decomposition (SVD) algorithm
did not converge (when computing the SVD of one of
the matrices |A-I  B2 |, |A-I  B1 |, D12 or D21).
|C1   D12|  |C2   D21|
= 6:  if the X-Riccati equation was not solved
successfully;
= 7:  if the matrix Im2 + B2'*X2*B2 is not positive
definite, or it is numerically singular (with
respect to the tolerance TOL);
= 8:  if the Y-Riccati equation was not solved
successfully;
= 9:  if the matrix Ip2 + C2*Y2*C2' is not positive
definite, or it is numerically singular (with
respect to the tolerance TOL);
=10:  if the matrix Im2 + DKHAT*D22 is singular, or its
estimated condition number is larger than or equal
to 1/TOL.

```
Method
```  The routine implements the formulas given in .

```
References
```   Zhou, K., Doyle, J.C., and Glover, K.
Robust and Optimal Control.
Prentice-Hall, Upper Saddle River, NJ, 1996.

 Petkov, P.Hr., Gu, D.W., and Konstantinov, M.M.
Fortran 77 routines for Hinf and H2 design of linear
discrete-time control systems.
Report 99-8, Department of Engineering, Leicester University,
April 1999.

```
Numerical Aspects
```  The accuracy of the result depends on the condition numbers of the
matrices which are to be inverted and on the condition numbers of
the matrix Riccati equations which are to be solved in the
computation of the controller. (The corresponding reciprocal
condition numbers are given in the output array RCOND.)

```
```  None
```
Example

Program Text

```*     SB10ED 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, LDB, LDC, LDD, LDAK, LDBK, LDCK, LDDK
PARAMETER        ( LDA = NMAX, LDB = NMAX, LDC = PMAX, LDD = PMAX,
\$                   LDAK = NMAX, LDBK = NMAX, LDCK = PMAX,
\$                   LDDK = PMAX )
INTEGER          LIWORK
PARAMETER        ( LIWORK = MAX( 2*MMAX, PMAX, 2*NMAX,
\$                                 NMAX*NMAX ) )
INTEGER          MPMX
PARAMETER        ( MPMX = MAX( MMAX, PMAX ) )
INTEGER          LDWORK
PARAMETER        ( LDWORK = 2*MPMX*( 3*MPMX + 2*NMAX ) +
\$                   MAX( ( NMAX + MPMX )*( NMAX + MPMX + 6 ),
\$                   MPMX*( MPMX + MAX( NMAX, MPMX, 5 ) + 1 ),
\$                   2*NMAX*NMAX + MAX( 14*NMAX*NMAX + 6*NMAX +
\$                   MAX( 14*NMAX + 23, 16*NMAX ),
\$                   MPMX*( NMAX + MPMX + MAX( MPMX, 3 ) ) ) ) )
*     .. Local Scalars ..
DOUBLE PRECISION TOL
INTEGER          I, INFO, J, M, N, NCON, NMEAS, NP
*     .. Local Arrays ..
LOGICAL          BWORK(2*NMAX)
INTEGER          IWORK(LIWORK)
DOUBLE PRECISION A(LDA,NMAX), AK(LDA,NMAX), B(LDB,MMAX),
\$                 BK(LDBK,MMAX), C(LDC,NMAX), CK(LDCK,NMAX),
\$                 D(LDD,MMAX), DK(LDDK,MMAX), DWORK(LDWORK),
\$                 RCOND( 8 )
*     .. External Subroutines ..
EXTERNAL         SB10ED
*     .. 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, NCON, NMEAS
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 IF ( NCON.LT.0 .OR. NCON.GT.MMAX ) THEN
WRITE ( NOUT, FMT = 99987 ) NCON
ELSE IF ( NMEAS.LT.0 .OR. NMEAS.GT.PMAX ) THEN
WRITE ( NOUT, FMT = 99986 ) NMEAS
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 = * ) TOL
CALL SB10ED( N, M, NP, NCON, NMEAS, A, LDA, B, LDB,
\$                C, LDC, D, LDD, AK, LDAK, BK, LDBK, CK, LDCK,
\$                DK, LDDK, RCOND, TOL, 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,NMEAS )
20       CONTINUE
WRITE ( NOUT, FMT = 99995 )
DO 30 I = 1, NCON
WRITE ( NOUT, FMT = 99992 ) ( CK(I,J), J = 1,N )
30       CONTINUE
WRITE ( NOUT, FMT = 99994 )
DO 40 I = 1, NCON
WRITE ( NOUT, FMT = 99992 ) ( DK(I,J), J = 1,NMEAS )
40       CONTINUE
WRITE( NOUT, FMT = 99993 )
WRITE( NOUT, FMT = 99991 ) ( RCOND(I), I = 1, 7 )
ELSE
WRITE( NOUT, FMT = 99998 ) INFO
END IF
END IF
STOP
*
99999 FORMAT (' SB10ED EXAMPLE PROGRAM RESULTS',/1X)
99998 FORMAT (/' INFO on exit from SB10ED =',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 (/' N is out of range.',/' N = ',I5)
99987 FORMAT (/' NCON is out of range.',/' NCON = ',I5)
99986 FORMAT (/' NMEAS is out of range.',/' NMEAS = ',I5)
END
```
Program Data
``` SB10ED EXAMPLE PROGRAM DATA
6     5     5     2     2
-0.7  0.0  0.3  0.0 -0.5 -0.1
-0.6  0.2 -0.4 -0.3  0.0  0.0
-0.5  0.7 -0.1  0.0  0.0 -0.8
-0.7  0.0  0.0 -0.5 -1.0  0.0
0.0  0.3  0.6 -0.9  0.1 -0.4
0.5 -0.8  0.0  0.0  0.2 -0.9
-1.0 -2.0 -2.0  1.0  0.0
1.0  0.0  1.0 -2.0  1.0
-3.0 -4.0  0.0  2.0 -2.0
1.0 -2.0  1.0  0.0 -1.0
0.0  1.0 -2.0  0.0  3.0
1.0  0.0  3.0 -1.0 -2.0
1.0 -1.0  2.0 -2.0  0.0 -3.0
-3.0  0.0  1.0 -1.0  1.0  0.0
0.0  2.0  0.0 -4.0  0.0 -2.0
1.0 -3.0  0.0  0.0  3.0  1.0
0.0  1.0 -2.0  1.0  0.0 -2.0
1.0 -1.0 -2.0  0.0  0.0
0.0  1.0  0.0  1.0  0.0
2.0 -1.0 -3.0  0.0  1.0
0.0  1.0  0.0  1.0 -1.0
0.0  0.0  1.0  2.0  1.0
0.00000001
```
Program Results
``` SB10ED EXAMPLE PROGRAM RESULTS

The controller state matrix AK is

-0.0551  -2.1891  -0.6607  -0.2532   0.6674  -1.0044
-1.0379   2.3804   0.5031   0.3960  -0.6605   1.2673
-0.0876  -2.1320  -0.4701  -1.1461   1.2927  -1.5116
-0.1358  -2.1237  -0.9560  -0.7144   0.6673  -0.7957
0.4900   0.0895   0.2634  -0.2354   0.1623  -0.2663
0.1672  -0.4163   0.2871  -0.1983   0.4944  -0.6967

The controller input matrix BK is

-0.5985  -0.5464
0.5285   0.6087
-0.7600  -0.4472
-0.7288  -0.6090
0.0532   0.0658
-0.0663   0.0059

The controller output matrix CK is

0.2500  -1.0200  -0.3371  -0.2733   0.2747  -0.4444
0.0654   0.2095   0.0632   0.2089  -0.1895   0.1834

The controller matrix DK is

-0.2181  -0.2070
0.1094   0.1159

The estimated condition numbers are

0.10000D+01  0.10000D+01  0.25207D+00  0.83985D-01  0.48628D-02
0.55015D-03  0.49886D+00
```