Nonlinear Control Sy.. - Free

8.2. DEFINITIONS 207
Definition 8.4 : A system H : X -a X is said to be strictly positive if there exists d > 0
such that
(u, Hu) > bI uI X + /.1 Vu E X. (8.12)
H is said to be positive if it satisfies (8.12) with b = 0.
The only difference between the notions of passivity and positivity (strict passivity
and strict positivity) is the lack of truncations in (8.12). As a consequence, the notions of
positivity and strict positivity apply to input-output stable systems exclusively. Notice that,
if the system H is not input-output stable, then the left-hand side of (8.12) is unbounded.
The following theorem shows that if a system is (i) causal and (ii) stable, then the
notions of positivity and passivity are entirely equivalent.
Theorem 8.1 Consider a system H : X --> X , and let H be causal. We have that
(i) H positive b H passive.
(ii) H strictly positive H strictly passive.
Proof: First assume that H satisfies (8.12) and consider an arbitrary input u E Xe . It
follows that UT E X, and by (8.12) we have that
but
(UT, HUT) ? 5U7' + 0
(UT, HUT) (UT, (HUT)T) by (8.9)
(UT, (Hu)T) since H is causal
(u, Hu)T by (8.9).
It follows that (u, HU)T > 6IIuTII2, and since u E Xe is arbitrary, we conclude that (8.12)
implies (8.11). For the converse, assume that H satisfies (8.11) and consider an arbitrary
input u E X. By (8.11), we have that
but
(u,Hu)T ? SIIuTIIX + a
(u, Hu)T = (UT, (Hu)T)
= (UT, (HUT)T)
= (UT, HUT)