## AB05SD

### Closed-loop system for an output feedback control law

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

Purpose

```  To construct for a given state space system (A,B,C,D) the closed-
loop system (Ac,Bc,Cc,Dc) corresponding to the output feedback
control law

u = alpha*F*y + v.

```
Specification
```      SUBROUTINE AB05SD( FBTYPE, JOBD, N, M, P, ALPHA, A, LDA, B, LDB,
\$                   C, LDC, D, LDD, F, LDF, RCOND, IWORK, DWORK,
\$                   LDWORK, INFO)
C     .. Scalar Arguments ..
CHARACTER         FBTYPE, JOBD
INTEGER           INFO, LDA, LDB, LDC, LDD, LDF, LDWORK, M, N, P
DOUBLE PRECISION  ALPHA, RCOND
C     .. Array Arguments ..
INTEGER           IWORK(*)
DOUBLE PRECISION  A(LDA,*), B(LDB,*), C(LDC,*), D(LDD,*),
\$                  DWORK(*), F(LDF,*)

```
Arguments

Mode Parameters

```  FBTYPE  CHARACTER*1
Specifies the type of the feedback law as follows:
= 'I':  Unitary output feedback (F = I);
= 'O':  General output feedback.

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.

```
Input/Output Parameters
```  N       (input) INTEGER
The number of state variables, i.e. the order of the
matrix A, the number of rows of B and the number of
columns of C.  N >= 0.

M       (input) INTEGER
The number of input variables, i.e. the number of columns
of matrices B and D, and the number of rows of F.  M >= 0.

P       (input) INTEGER
The number of output variables, i.e. the number of rows of
matrices C and D, and the number of columns of F.  P >= 0
and P = M if FBTYPE = 'I'.

ALPHA   (input) DOUBLE PRECISION
The coefficient alpha in the output feedback law.

A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
On entry, the leading N-by-N part of this array must
contain the system state transition matrix A.
On exit, the leading N-by-N part of this array contains
the state matrix Ac of the closed-loop system.

LDA     INTEGER
The leading dimension of 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 input matrix Bc of the closed-loop system.

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

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

LDC     INTEGER
The leading dimension of array C.
LDC >= MAX(1,P) if N > 0.
LDC >= 1 if N = 0.

D       (input/output) DOUBLE PRECISION array, dimension (LDD,M)
On entry, the leading P-by-M part of this array must
contain the system direct input/output transmission
matrix D.
On exit, if JOBD = 'D', the leading P-by-M part of this
array contains the direct input/output transmission
matrix Dc of the closed-loop system.
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) DOUBLE PRECISION array, dimension (LDF,P)
If FBTYPE = 'O', the leading M-by-P part of this array
must contain the output feedback matrix F.
If FBTYPE = 'I', then the feedback matrix is assumed to be
an M x M order identity matrix.
The array F is not referenced if FBTYPE = 'I' or
ALPHA = 0.

LDF     INTEGER
The leading dimension of array F.
LDF >= MAX(1,M) if FBTYPE = 'O' and ALPHA <> 0.
LDF >= 1 if FBTYPE = 'I' or ALPHA = 0.

RCOND   (output) DOUBLE PRECISION
The reciprocal condition number of the matrix
I - alpha*D*F.

```
Workspace
```  IWORK   INTEGER array, dimension (LIWORK)
LIWORK >= MAX(1,2*P) if JOBD = 'D'.
LIWORK >= 1 if JOBD = 'Z'.
IWORK is not referenced if JOBD = 'Z'.

DWORK   DOUBLE PRECISION array, dimension (LDWORK)

LDWORK  INTEGER
The length of the array DWORK.
LDWORK >= wspace, where
wspace = MAX( 1, M, P*P + 4*P ) if JOBD = 'D',
wspace = MAX( 1, M ) if JOBD = 'Z'.
For best performance, LDWORK >= MAX( wspace, N*M, N*P ).

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value;
= 1:  if the matrix I - alpha*D*F is numerically singular.

```
Method
```  The matrices of the closed-loop system have the expressions:

Ac = A + alpha*B*F*E*C,  Bc = B + alpha*B*F*E*D,
Cc = E*C,                Dc = E*D,

where E = (I - alpha*D*F)**-1.

```
Numerical Aspects
```  The accuracy of computations basically depends on the conditioning
of the matrix I - alpha*D*F.  If RCOND is very small, it is likely
that the computed results are inaccurate.

```
```  None
```
Example

Program Text

```  None
```
Program Data
```  None
```
Program Results
```  None
```