Nonlinear Control Sy.. - Free

236 CHAPTER 9. DISSIPATIVITY
which implies that (9.22)-(9.23) are satisfied if and only if
ATP+PA < 0
BTP = CT.
Therefore, for passive systems, Theorem 9.2 can be considered as a nonlinear version of the
Kalman-Yakubovich lemma discussed in Chapter 8.
Strictly output passive systems: Now consider the strictly output passivity supply rate
w(u,y) = uTy - eyTy (i.e., Q = -JI, R = 0, and S = 2I in (9.6)), and assume once
again that j (x) = 0. In this case Theorem 9.2 states that the nonlinear system z/) is strictly
output-passive if and only if
or, equivalently,
a7 f(x) = -ehT(x)h(x) - LT(x)L(x)
a f(x) < -ehT(x)h(x)
ao g
hT (x).
(9.24)
(9.25)
Strictly Passive systems: Finally consider the strict passivity supply rate w(u, y) =
uT y - 5UT u (i.e., Q = 0, R = -8I, and S = 2I in (9.6)), and assume once again that
j(x) = 0. In this case Theorem 9.2 states that the nonlinear system t,b is strictly passive if
and only if
with
(x) = -LT (x)L(x)
axf
a
gT (ax) = h(x) - 2WTL
R=R=WTW=-8I
which can never be satisfied since WTW > 0 and 8 > 0. It then follows that no system of
the form (9.13), with j = 0, can be strictly passive.