## SB10PD

### Normalization of a system for H-infinity controller design

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

Purpose

```  To reduce the matrices D12 and D21 of the linear time-invariant
system

| A  | B1  B2  |   | A | B |
P = |----|---------| = |---|---|
| C1 | D11 D12 |   | C | D |
| C2 | D21 D22 |

to unit diagonal form, to transform the matrices B, C, and D11 to
satisfy the formulas in the computation of an H2 and H-infinity
(sub)optimal controllers and to check the rank conditions.

```
Specification
```      SUBROUTINE SB10PD( N, M, NP, NCON, NMEAS, A, LDA, B, LDB, C, LDC,
\$                   D, LDD, TU, LDTU, TY, LDTY, RCOND, TOL, DWORK,
\$                   LDWORK, INFO )
C     .. Scalar Arguments ..
INTEGER            INFO, LDA, LDB, LDC, LDD, LDTU, LDTY, LDWORK,
\$                   M, N, NCON, NMEAS, NP
DOUBLE PRECISION   TOL
C     .. Array Arguments ..
DOUBLE PRECISION   A( LDA, * ), B( LDB, * ), C( LDC, * ),
\$                   D( LDD, * ), DWORK( * ), RCOND( 2 ),
\$                   TU( LDTU, * ), TY( LDTY, * )

```
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) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
system state matrix A.

LDA     INTEGER
The leading dimension of the 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 system input matrix B.
On exit, the leading N-by-M part of this array contains
the transformed system input matrix B.

LDB     INTEGER
The leading dimension of the array B.  LDB >= max(1,N).

C       (input/output) DOUBLE PRECISION array, dimension (LDC,N)
On entry, the leading NP-by-N part of this array must
contain the system output matrix C.
On exit, the leading NP-by-N part of this array contains
the transformed system output matrix C.

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

D       (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading NP-by-M part of this array must
contain the system input/output matrix D. The
NMEAS-by-NCON trailing submatrix D22 is not referenced.
On exit, the leading (NP-NMEAS)-by-(M-NCON) part of this
array contains the transformed submatrix D11.
The transformed submatrices D12 = [ 0  Im2 ]' and
D21 = [ 0  Inp2 ] are not stored. The corresponding part
of this array contains no useful information.

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

TU      (output) DOUBLE PRECISION array, dimension (LDTU,M2)
The leading M2-by-M2 part of this array contains the
control transformation matrix TU.

LDTU    INTEGER
The leading dimension of the array TU.  LDTU >= max(1,M2).

TY      (output) DOUBLE PRECISION array, dimension (LDTY,NP2)
The leading NP2-by-NP2 part of this array contains the
measurement transformation matrix TY.

LDTY    INTEGER
The leading dimension of the array TY.
LDTY >= max(1,NP2).

RCOND   (output) DOUBLE PRECISION array, dimension (2)
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 is set even if INFO = 3 or INFO = 4; if INFO = 3,
then RCOND(2) was not computed, but it is set to 0.

```
Tolerances
```  TOL     DOUBLE PRECISION
Tolerance used for controlling the accuracy of the applied
transformations. Transformation matrices TU and TY whose
reciprocal condition numbers are less than TOL are not
allowed. If TOL <= 0, then a default value equal to
sqrt(EPS) is used, where EPS is the relative machine
precision.

```
Workspace
```  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 >= MAX(1,LW1,LW2,LW3,LW4), 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),
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
MAX(1,(N+Q)*(N+Q+6),Q*(Q+MAX(N,Q,5)+1).

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value;
= 1:  if the matrix | A   B2  | had not full column rank
| C1  D12 |
in respect to the tolerance EPS;
= 2:  if the matrix | A   B1  | had not full row rank in
| C2  D21 |
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   B2 |, |A   B1 |, D12 or D21).
|C1  D12|  |C2  D21|

```
Method
```  The routine performs the transformations described in .

```
References
```   Glover, K. and Doyle, J.C.
State-space formulae for all stabilizing controllers that
satisfy an Hinf norm bound and relations to risk sensitivity.
Systems and Control Letters, vol. 11, pp. 167-172, 1988.

 Balas, G.J., Doyle, J.C., Glover, K., Packard, A., and
Smith, R.
mu-Analysis and Synthesis Toolbox.
The MathWorks Inc., Natick, Mass., 1995.

```
Numerical Aspects
```  The precision of the transformations can be controlled by the
condition numbers of the matrices TU and TY as given by the
values of RCOND(1) and RCOND(2), respectively. An error return
with INFO = 3 or INFO = 4 will be obtained if the condition
number of TU or TY, respectively, would exceed 1/TOL.

```
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Example

Program Text

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Program Data
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Program Results
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