Nonlinear Control Sy.. - Free

8.4. STABILITY OF FEEDBACK INTERCONNECTIONS 211
ul(t) + el(t)
H1
H2
e2(t)
yi(t)
Figure 8.5: The Feedback System S1.
(a) H is passive if and only if the gain of S is at most 1, that is, S is such that
I(Sx)TI x < II-TIIX Vx E Xe ,VT E Xe. (8.17)
(b) H is strictly passive and has finite gain if and only if the gain of S is less than 1.
Proof: See the Appendix.
8.4 Stability of Feedback Interconnections
In this section we exploit the concept of passivity in the stability analysis of feedback
interconnections. To simplify our proofs, we assume without loss of generality that the
systems are initially relaxed, and so the constant )3 in Definitions 8.2 and 8.3 is identically
zero.
Our first result consists of the simplest form of the passivity theorem. The simplicity
of the theorem stems from considering a feedback system with one single input u1, as shown
in Figure 8.5 (u2 = 0 in the feedback system used in Chapter 6).
Theorem 8.4 : Let H1, H2 : Xe 4 Xe and consider the feedback interconnection defined
by the following equations:
ei = ul - H2e2 (8.18)
yi = H1e1.
(8.19)