IB01RD

Estimating initial state, given A, B, C, D, and input-output trajectories

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

Purpose

```  To estimate the initial state of a linear time-invariant (LTI)
discrete-time system, given the system matrices  (A,B,C,D)  and
the input and output trajectories of the system. The model
structure is :

x(k+1) = Ax(k) + Bu(k),   k >= 0,
y(k)   = Cx(k) + Du(k),

where  x(k)  is the  n-dimensional state vector (at time k),
u(k)  is the  m-dimensional input vector,
y(k)  is the  l-dimensional output vector,
and  A, B, C, and D  are real matrices of appropriate dimensions.
Matrix  A  is assumed to be in a real Schur form.

```
Specification
```      SUBROUTINE IB01RD( JOB, N, M, L, NSMP, A, LDA, B, LDB, C, LDC, D,
\$                   LDD, U, LDU, Y, LDY, X0, TOL, IWORK, DWORK,
\$                   LDWORK, IWARN, INFO )
C     .. Scalar Arguments ..
DOUBLE PRECISION   TOL
INTEGER            INFO, IWARN, L, LDA, LDB, LDC, LDD, LDU,
\$                   LDWORK, LDY, M, N, NSMP
CHARACTER          JOB
C     .. Array Arguments ..
DOUBLE PRECISION   A(LDA, *), B(LDB, *), C(LDC, *), D(LDD, *),
\$                   DWORK(*),  U(LDU, *), X0(*), Y(LDY, *)
INTEGER            IWORK(*)

```
Arguments

Mode Parameters

```  JOB     CHARACTER*1
Specifies whether or not the matrix D is zero, as follows:
= 'Z':  the matrix  D  is zero;
= 'N':  the matrix  D  is not zero.

```
Input/Output Parameters
```  N       (input) INTEGER
The order of the system.  N >= 0.

M       (input) INTEGER
The number of system inputs.  M >= 0.

L       (input) INTEGER
The number of system outputs.  L > 0.

NSMP    (input) INTEGER
The number of rows of matrices  U  and  Y  (number of
samples used,  t).  NSMP >= N.

A       (input) DOUBLE PRECISION array, dimension (LDA,N)
The leading N-by-N part of this array must contain the
system state matrix  A  in a real Schur form.

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  (corresponding to the real Schur
form of  A).
If  N = 0  or  M = 0,  this array is not referenced.

LDB     INTEGER
The leading dimension of the array B.
LDB >= N,  if  N > 0  and  M > 0;
LDB >= 1,  if  N = 0  or   M = 0.

C       (input) DOUBLE PRECISION array, dimension (LDC,N)
The leading L-by-N part of this array must contain the
system output matrix  C  (corresponding to the real Schur
form of  A).

LDC     INTEGER
The leading dimension of the array C.  LDC >= L.

D       (input) DOUBLE PRECISION array, dimension (LDD,M)
The leading L-by-M part of this array must contain the
system input-output matrix.
If  M = 0  or  JOB = 'Z',  this array is not referenced.

LDD     INTEGER
The leading dimension of the array D.
LDD >= L,  if  M > 0  and  JOB = 'N';
LDD >= 1,  if  M = 0  or   JOB = 'Z'.

U       (input) DOUBLE PRECISION array, dimension (LDU,M)
If  M > 0,  the leading NSMP-by-M part of this array must
contain the t-by-m input-data sequence matrix  U,
U = [u_1 u_2 ... u_m].  Column  j  of  U  contains the
NSMP  values of the j-th input component for consecutive
time increments.
If M = 0, this array is not referenced.

LDU     INTEGER
The leading dimension of the array U.
LDU >= MAX(1,NSMP),  if M > 0;
LDU >= 1,            if M = 0.

Y       (input) DOUBLE PRECISION array, dimension (LDY,L)
The leading NSMP-by-L part of this array must contain the
t-by-l output-data sequence matrix  Y,
Y = [y_1 y_2 ... y_l].  Column  j  of  Y  contains the
NSMP  values of the j-th output component for consecutive
time increments.

LDY     INTEGER
The leading dimension of the array Y.  LDY >= MAX(1,NSMP).

X0      (output) DOUBLE PRECISION array, dimension (N)
The estimated initial state of the system,  x(0).

```
Tolerances
```  TOL     DOUBLE PRECISION
The tolerance to be used for estimating the rank of
matrices. If the user sets  TOL > 0,  then the given value
of  TOL  is used as a lower bound for the reciprocal
condition number;  a matrix whose estimated condition
number is less than  1/TOL  is considered to be of full
rank.  If the user sets  TOL <= 0,  then  EPS  is used
instead, where  EPS  is the relative machine precision
(see LAPACK Library routine DLAMCH).  TOL <= 1.

```
Workspace
```  IWORK   INTEGER array, dimension (N)

DWORK   DOUBLE PRECISION array, dimension (LDWORK)
On exit, if  INFO = 0,  DWORK(1) returns the optimal value
of LDWORK and  DWORK(2)  contains the reciprocal condition
number of the triangular factor of the QR factorization of
the matrix  Gamma  (see METHOD).
On exit, if  INFO = -22,  DWORK(1)  returns the minimum
value of LDWORK.

LDWORK  INTEGER
The length of the array DWORK.
LDWORK >= max( 2, min( LDW1, LDW2 ) ),  where
LDW1 = t*L*(N + 1) + 2*N + max( 2*N*N, 4*N ),
LDW2 =   N*(N + 1) + 2*N +
max( q*(N + 1) + 2*N*N + L*N, 4*N ),
q = N*L.
For good performance,  LDWORK  should be larger.
If  LDWORK >= LDW1,  then standard QR factorization of
the matrix  Gamma  (see METHOD) is used. Otherwise, the
QR factorization is computed sequentially by performing
NCYCLE  cycles, each cycle (except possibly the last one)
processing  s  samples, where  s  is chosen by equating
LDWORK  to  LDW2,  for  q  replaced by  s*L.
The computational effort may increase and the accuracy may
decrease with the decrease of  s.  Recommended value is
LDRWRK = LDW1,  assuming a large enough cache size, to
also accommodate  A, B, C, D, U,  and  Y.

```
Warning Indicator
```  IWARN   INTEGER
= 0:  no warning;
= 4:  the least squares problem to be solved has a
rank-deficient coefficient matrix.

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value;
= 2:  the singular value decomposition (SVD) algorithm did
not converge.

```
Method
```  An extension and refinement of the method in [1] is used.
Specifically, the output y0(k) of the system for zero initial
state is computed for k = 0, 1, ...,  t-1 using the given model.
Then the following least squares problem is solved for x(0)

(     C     )            (   y(0) - y0(0)   )
(    C*A    )            (   y(1) - y0(1)   )
Gamma * x(0)  =  (     :     ) * x(0)  =  (        :         ).
(     :     )            (        :         )
( C*A^(t-1) )            ( y(t-1) - y0(t-1) )

The coefficient matrix  Gamma  is evaluated using powers of A with
exponents 2^k. The QR decomposition of this matrix is computed.
If its triangular factor  R  is too ill conditioned, then singular
value decomposition of  R  is used.

If the coefficient matrix cannot be stored in the workspace (i.e.,
LDWORK < LDW1),  the QR decomposition is computed sequentially.

```
References
```  [1] Verhaegen M., and Varga, A.
Some Experience with the MOESP Class of Subspace Model
Identification Methods in Identifying the BO105 Helicopter.
Report TR R165-94, DLR Oberpfaffenhofen, 1994.

```
Numerical Aspects
```  The implemented method is numerically stable.

```
```  None
```
Example

Program Text

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