qLet P and Q be the controllability
and observability Gramians,
•AP+PA*+BB*=0, A*Q+QA+C*C=0
• Suppose P=Q=S=diag(s1, s2,…, sn). Then the state space realization is called internally balanced
realization and s1 ³ s2 ³ … ³sn ³0, are called the Hankel singular values of the system.
qTwo other closely related
realizations are called
q
input
normal realization with P = I and Q = S2
q and
q output
normal realization with P = S2 and Q = I.
q Both
realizations can be obtained easily from the balanced realization by a suitable scaling on
the states.