10

Computing Balanced Realization

qIn the special case
where is a minimal
realization, a balanced realization can be obtained through the
following simplified procedure:

1. Compute *P*>0
and *Q*>0.

2. Find a matrix *R*
such that *P
=R*R.
*

3. Diagonalize *RQR** to get *RQR*=U
**S**2**U*.*

4. Let *T **–1**=R*U **S**-1/2**. *Then *T PT*=(T*)**–1**QT **–1**=**S*
and is balanced.

qIn general, let *P* and *Q* be two positive
semidefinite matrices. Then there exists a nonsingular matrix *T* such that

respectively with *S**1**, **S**2**, **S**3* diagonal and positive definite.