Nonlinear Control Sy.. - Free

208 CHAPTER 8. PASSIVITY
Thus (UT, HUT) > IIUTII2 + /3, which is valid for all T E R+. Moreover, since u E X and
H : X - X, we can take limits as T -* oo to obtain
(u, Hu) > 8IIuIIX + Q
Thus we conclude that (8.11) implies (8.12) and the second assertion of the theorem is
proved. Part (i) is immediately obvious assuming that b = 0. This completes the proof.
8.3 Interconnections of Passivity Systems
In many occasions it is important to study the properties of combinations of passive systems.
The following Theorem considers two important cases
Theorem 8.2 Consider a finite number of systems Hi : Xe - Xe , i = 1, , n. We have
(i) If all of the systems Hi, i = 1, , n are passive, then the system H : Xe + Xe ,
defined by (see Figure 8.3)
(8.13)
is passive.
(ii) If all the systems Hi, i = 1, , n are passive, and at least one of them is strictly
passive, then the system H defined by equation (8.13) is strictly passive.
(iii) If the systems Hi, i = 1, 2 are passive and the feedback interconnection defined by the
equations (Figure 8.4)
e = u-H2y (8.14)
y = Hie (8.15)
is well defined (i.e., e(t) E Xe and is uniquely determined for each u(t) E Xe), then,
the mapping from u into y defined by equations (8.14)-(8.15) is passive.
Proof of Theorem 8.2
Proof of (i): We have
(x,(Hl+...+Hn)x)T = (x,Hlx+...+Hnx)T
(x+ Hlx)T + ... + (x+ Hnx)T
deJ N.