9

Balanced Realization

qLet *P* and *Q* be the controllability
and observability Gramians,

Suppose *P=Q=**S**=diag(**s**1**, **s**2**,…, **s**n**). *Then the state space realization is called **internally balanced
realization** and *s**1 *³* **s**2 *³* **… *³*s**n* ³0, are called the **Hankel singular values** of the system.

qTwo other closely related
realizations are called

and

Both
realizations can be obtained easily from the balanced realization by a suitable scaling on
the states.