Nonlinear Control Sy.. - Free

216 CHAPTER 8. PASSIVITY
(i) H is passive if and only if [H(,w)] + H(3w)*] > 0 Vw E R.
(ii) H is strictly passive if and only if 3b > 0 such that
Amin[1 (3w)] + H(3w)'] > b Vw E ]R
It is important to notice that Theorem 8.7 was proved for systems whose impulse
response is in the algebra A. In particular, for finite-dimensional systems, this algebra
consists of systems with all of their poles in the open left half of the complex plane. Thus,
Theorem 8.7 says nothing about whether a system with transfer function
as
H(s) = w2 a> 0, w o > 0
(8.24)
s2
0
is passive. This transfer function, in turn, is the building block of a very important class
of system to be described later. Our next theorem shows that the class of systems with a
transfer function of the form (8.24) is indeed passive, regardless of the particular value of
a and w.
Theorem 8.9 Consider the system H : Gee -> G2e defined by its transfer function
H(s) = s2
Under these conditions, H is passive.
as
w20 Proof: By Theorem 8.3, H is passive if and only if
but for the given H(s)
S(s) _
SOW) _
1-H
a > 0, w0 > 0.
1+H 00